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Theorem eupath2 26507
 Description: The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
Hypotheses
Ref Expression
eupath2.1 (𝜑𝐸 Fn 𝐴)
eupath2.3 (𝜑𝐹(𝑉 EulPaths 𝐸)𝑃)
Assertion
Ref Expression
eupath2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))
Distinct variable groups:   𝑥,𝐸   𝑥,𝐹   𝑥,𝑉   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝑃(𝑥)

Proof of Theorem eupath2
Dummy variables 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eupath2.3 . . . . . . . . . . 11 (𝜑𝐹(𝑉 EulPaths 𝐸)𝑃)
2 eupath2.1 . . . . . . . . . . 11 (𝜑𝐸 Fn 𝐴)
3 eupaf1o 26497 . . . . . . . . . . 11 ((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → 𝐹:(1...(#‘𝐹))–1-1-onto𝐴)
41, 2, 3syl2anc 691 . . . . . . . . . 10 (𝜑𝐹:(1...(#‘𝐹))–1-1-onto𝐴)
5 f1ofo 6057 . . . . . . . . . 10 (𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝐹:(1...(#‘𝐹))–onto𝐴)
6 foima 6033 . . . . . . . . . 10 (𝐹:(1...(#‘𝐹))–onto𝐴 → (𝐹 “ (1...(#‘𝐹))) = 𝐴)
74, 5, 63syl 18 . . . . . . . . 9 (𝜑 → (𝐹 “ (1...(#‘𝐹))) = 𝐴)
87reseq2d 5317 . . . . . . . 8 (𝜑 → (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))) = (𝐸𝐴))
9 fnresdm 5914 . . . . . . . . 9 (𝐸 Fn 𝐴 → (𝐸𝐴) = 𝐸)
102, 9syl 17 . . . . . . . 8 (𝜑 → (𝐸𝐴) = 𝐸)
118, 10eqtrd 2644 . . . . . . 7 (𝜑 → (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))) = 𝐸)
1211oveq2d 6565 . . . . . 6 (𝜑 → (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹))))) = (𝑉 VDeg 𝐸))
1312fveq1d 6105 . . . . 5 (𝜑 → ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑥))
1413breq2d 4595 . . . 4 (𝜑 → (2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥) ↔ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)))
1514notbid 307 . . 3 (𝜑 → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥) ↔ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)))
1615rabbidv 3164 . 2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)})
17 eupacl 26496 . . . . 5 (𝐹(𝑉 EulPaths 𝐸)𝑃 → (#‘𝐹) ∈ ℕ0)
18 nn0re 11178 . . . . 5 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℝ)
191, 17, 183syl 18 . . . 4 (𝜑 → (#‘𝐹) ∈ ℝ)
2019leidd 10473 . . 3 (𝜑 → (#‘𝐹) ≤ (#‘𝐹))
211, 17syl 17 . . . 4 (𝜑 → (#‘𝐹) ∈ ℕ0)
22 breq1 4586 . . . . . . 7 (𝑚 = 0 → (𝑚 ≤ (#‘𝐹) ↔ 0 ≤ (#‘𝐹)))
23 oveq2 6557 . . . . . . . . . . . . . . . . . 18 (𝑚 = 0 → (1...𝑚) = (1...0))
24 fz10 12233 . . . . . . . . . . . . . . . . . 18 (1...0) = ∅
2523, 24syl6eq 2660 . . . . . . . . . . . . . . . . 17 (𝑚 = 0 → (1...𝑚) = ∅)
2625imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (𝐹 “ (1...𝑚)) = (𝐹 “ ∅))
27 ima0 5400 . . . . . . . . . . . . . . . 16 (𝐹 “ ∅) = ∅
2826, 27syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝐹 “ (1...𝑚)) = ∅)
2928reseq2d 5317 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝐸 ↾ (𝐹 “ (1...𝑚))) = (𝐸 ↾ ∅))
30 res0 5321 . . . . . . . . . . . . . 14 (𝐸 ↾ ∅) = ∅
3129, 30syl6eq 2660 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝐸 ↾ (𝐹 “ (1...𝑚))) = ∅)
3231oveq2d 6565 . . . . . . . . . . . 12 (𝑚 = 0 → (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚)))) = (𝑉 VDeg ∅))
3332fveq1d 6105 . . . . . . . . . . 11 (𝑚 = 0 → ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) = ((𝑉 VDeg ∅)‘𝑥))
3433breq2d 4595 . . . . . . . . . 10 (𝑚 = 0 → (2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)))
3534notbid 307 . . . . . . . . 9 (𝑚 = 0 → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)))
3635rabbidv 3164 . . . . . . . 8 (𝑚 = 0 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)})
37 fveq2 6103 . . . . . . . . . 10 (𝑚 = 0 → (𝑃𝑚) = (𝑃‘0))
3837eqcomd 2616 . . . . . . . . 9 (𝑚 = 0 → (𝑃‘0) = (𝑃𝑚))
3938iftrued 4044 . . . . . . . 8 (𝑚 = 0 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = ∅)
4036, 39eqeq12d 2625 . . . . . . 7 (𝑚 = 0 → ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)} = ∅))
4122, 40imbi12d 333 . . . . . 6 (𝑚 = 0 → ((𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ (0 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)} = ∅)))
4241imbi2d 329 . . . . 5 (𝑚 = 0 → ((𝜑 → (𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → (0 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)} = ∅))))
43 breq1 4586 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ≤ (#‘𝐹) ↔ 𝑛 ≤ (#‘𝐹)))
44 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
4544imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝐹 “ (1...𝑚)) = (𝐹 “ (1...𝑛)))
4645reseq2d 5317 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝐸 ↾ (𝐹 “ (1...𝑚))) = (𝐸 ↾ (𝐹 “ (1...𝑛))))
4746oveq2d 6565 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚)))) = (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛)))))
4847fveq1d 6105 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) = ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥))
4948breq2d 4595 . . . . . . . . . 10 (𝑚 = 𝑛 → (2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)))
5049notbid 307 . . . . . . . . 9 (𝑚 = 𝑛 → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)))
5150rabbidv 3164 . . . . . . . 8 (𝑚 = 𝑛 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)})
52 fveq2 6103 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑃𝑚) = (𝑃𝑛))
5352eqeq2d 2620 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃𝑛)))
5452preq2d 4219 . . . . . . . . 9 (𝑚 = 𝑛 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃𝑛)})
5553, 54ifbieq2d 4061 . . . . . . . 8 (𝑚 = 𝑛 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))
5651, 55eqeq12d 2625 . . . . . . 7 (𝑚 = 𝑛 → ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})))
5743, 56imbi12d 333 . . . . . 6 (𝑚 = 𝑛 → ((𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ (𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))))
5857imbi2d 329 . . . . 5 (𝑚 = 𝑛 → ((𝜑 → (𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → (𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})))))
59 breq1 4586 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ≤ (#‘𝐹) ↔ (𝑛 + 1) ≤ (#‘𝐹)))
60 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
6160imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (𝐹 “ (1...𝑚)) = (𝐹 “ (1...(𝑛 + 1))))
6261reseq2d 5317 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → (𝐸 ↾ (𝐹 “ (1...𝑚))) = (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))
6362oveq2d 6565 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚)))) = (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1))))))
6463fveq1d 6105 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) = ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥))
6564breq2d 4595 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)))
6665notbid 307 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)))
6766rabbidv 3164 . . . . . . . 8 (𝑚 = (𝑛 + 1) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)})
68 fveq2 6103 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝑃𝑚) = (𝑃‘(𝑛 + 1)))
6968eqeq2d 2620 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
7068preq2d 4219 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
7169, 70ifbieq2d 4061 . . . . . . . 8 (𝑚 = (𝑛 + 1) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
7267, 71eqeq12d 2625 . . . . . . 7 (𝑚 = (𝑛 + 1) → ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
7359, 72imbi12d 333 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
7473imbi2d 329 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝜑 → (𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))))
75 breq1 4586 . . . . . . 7 (𝑚 = (#‘𝐹) → (𝑚 ≤ (#‘𝐹) ↔ (#‘𝐹) ≤ (#‘𝐹)))
76 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑚 = (#‘𝐹) → (1...𝑚) = (1...(#‘𝐹)))
7776imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑚 = (#‘𝐹) → (𝐹 “ (1...𝑚)) = (𝐹 “ (1...(#‘𝐹))))
7877reseq2d 5317 . . . . . . . . . . . . 13 (𝑚 = (#‘𝐹) → (𝐸 ↾ (𝐹 “ (1...𝑚))) = (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))
7978oveq2d 6565 . . . . . . . . . . . 12 (𝑚 = (#‘𝐹) → (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚)))) = (𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹))))))
8079fveq1d 6105 . . . . . . . . . . 11 (𝑚 = (#‘𝐹) → ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) = ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥))
8180breq2d 4595 . . . . . . . . . 10 (𝑚 = (#‘𝐹) → (2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)))
8281notbid 307 . . . . . . . . 9 (𝑚 = (#‘𝐹) → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥) ↔ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)))
8382rabbidv 3164 . . . . . . . 8 (𝑚 = (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)})
84 fveq2 6103 . . . . . . . . . 10 (𝑚 = (#‘𝐹) → (𝑃𝑚) = (𝑃‘(#‘𝐹)))
8584eqeq2d 2620 . . . . . . . . 9 (𝑚 = (#‘𝐹) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(#‘𝐹))))
8684preq2d 4219 . . . . . . . . 9 (𝑚 = (#‘𝐹) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(#‘𝐹))})
8785, 86ifbieq2d 4061 . . . . . . . 8 (𝑚 = (#‘𝐹) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))
8883, 87eqeq12d 2625 . . . . . . 7 (𝑚 = (#‘𝐹) → ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})))
8975, 88imbi12d 333 . . . . . 6 (𝑚 = (#‘𝐹) → ((𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ ((#‘𝐹) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))))
9089imbi2d 329 . . . . 5 (𝑚 = (#‘𝐹) → ((𝜑 → (𝑚 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑚))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → ((#‘𝐹) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})))))
91 2z 11286 . . . . . . . . . . 11 2 ∈ ℤ
92 dvds0 14835 . . . . . . . . . . 11 (2 ∈ ℤ → 2 ∥ 0)
9391, 92ax-mp 5 . . . . . . . . . 10 2 ∥ 0
94 eupagra 26493 . . . . . . . . . . . 12 (𝐹(𝑉 EulPaths 𝐸)𝑃𝑉 UMGrph 𝐸)
95 relumgra 25843 . . . . . . . . . . . . 13 Rel UMGrph
9695brrelexi 5082 . . . . . . . . . . . 12 (𝑉 UMGrph 𝐸𝑉 ∈ V)
971, 94, 963syl 18 . . . . . . . . . . 11 (𝜑𝑉 ∈ V)
98 vdgr0 26427 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝑥𝑉) → ((𝑉 VDeg ∅)‘𝑥) = 0)
9997, 98sylan 487 . . . . . . . . . 10 ((𝜑𝑥𝑉) → ((𝑉 VDeg ∅)‘𝑥) = 0)
10093, 99syl5breqr 4621 . . . . . . . . 9 ((𝜑𝑥𝑉) → 2 ∥ ((𝑉 VDeg ∅)‘𝑥))
101 notnot 135 . . . . . . . . 9 (2 ∥ ((𝑉 VDeg ∅)‘𝑥) → ¬ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥))
102100, 101syl 17 . . . . . . . 8 ((𝜑𝑥𝑉) → ¬ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥))
103102ralrimiva 2949 . . . . . . 7 (𝜑 → ∀𝑥𝑉 ¬ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥))
104 rabeq0 3911 . . . . . . 7 ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)} = ∅ ↔ ∀𝑥𝑉 ¬ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥))
105103, 104sylibr 223 . . . . . 6 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)} = ∅)
106105a1d 25 . . . . 5 (𝜑 → (0 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg ∅)‘𝑥)} = ∅))
107 nn0re 11178 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
108107adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
109108lep1d 10834 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ≤ (𝑛 + 1))
110 peano2re 10088 . . . . . . . . . . . 12 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
111108, 110syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℝ)
11219adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (#‘𝐹) ∈ ℝ)
113 letr 10010 . . . . . . . . . . 11 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (#‘𝐹) ∈ ℝ) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (#‘𝐹)) → 𝑛 ≤ (#‘𝐹)))
114108, 111, 112, 113syl3anc 1318 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (#‘𝐹)) → 𝑛 ≤ (#‘𝐹)))
115109, 114mpand 707 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (#‘𝐹) → 𝑛 ≤ (#‘𝐹)))
116115imim1d 80 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))))
117 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥) = ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦))
118117breq2d 4595 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥) ↔ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦)))
119118notbid 307 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥) ↔ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦)))
120119elrab 3331 . . . . . . . . . . . 12 (𝑦 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} ↔ (𝑦𝑉 ∧ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦)))
1212ad3antrrr 762 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝐸 Fn 𝐴)
1221ad3antrrr 762 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝐹(𝑉 EulPaths 𝐸)𝑃)
123 simpllr 795 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝑛 ∈ ℕ0)
124 simplrl 796 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → (𝑛 + 1) ≤ (#‘𝐹))
125 simpr 476 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝑦𝑉)
126 simplrr 797 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))
127121, 122, 123, 124, 125, 126eupath2lem3 26506 . . . . . . . . . . . . . 14 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦) ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
128127pm5.32da 671 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → ((𝑦𝑉 ∧ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦)) ↔ (𝑦𝑉𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
129 0elpw 4760 . . . . . . . . . . . . . . . . 17 ∅ ∈ 𝒫 𝑉
130 eupapf 26499 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹(𝑉 EulPaths 𝐸)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
1311, 130syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)
132131ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → 𝑃:(0...(#‘𝐹))⟶𝑉)
13321ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (#‘𝐹) ∈ ℕ0)
134 nn0uz 11598 . . . . . . . . . . . . . . . . . . . . . 22 0 = (ℤ‘0)
135133, 134syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (#‘𝐹) ∈ (ℤ‘0))
136 eluzfz1 12219 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐹) ∈ (ℤ‘0) → 0 ∈ (0...(#‘𝐹)))
137135, 136syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → 0 ∈ (0...(#‘𝐹)))
138132, 137ffvelrnd 6268 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑃‘0) ∈ 𝑉)
139 simprl 790 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ≤ (#‘𝐹))
140 peano2nn0 11210 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
141140ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ∈ ℕ0)
142141, 134syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ∈ (ℤ‘0))
143133nn0zd 11356 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (#‘𝐹) ∈ ℤ)
144 elfz5 12205 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 + 1) ∈ (ℤ‘0) ∧ (#‘𝐹) ∈ ℤ) → ((𝑛 + 1) ∈ (0...(#‘𝐹)) ↔ (𝑛 + 1) ≤ (#‘𝐹)))
145142, 143, 144syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → ((𝑛 + 1) ∈ (0...(#‘𝐹)) ↔ (𝑛 + 1) ≤ (#‘𝐹)))
146139, 145mpbird 246 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ∈ (0...(#‘𝐹)))
147132, 146ffvelrnd 6268 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑃‘(𝑛 + 1)) ∈ 𝑉)
148 prssi 4293 . . . . . . . . . . . . . . . . . . 19 (((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(𝑛 + 1)) ∈ 𝑉) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
149138, 147, 148syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
150 prex 4836 . . . . . . . . . . . . . . . . . . 19 {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
151150elpw 4114 . . . . . . . . . . . . . . . . . 18 ({(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
152149, 151sylibr 223 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉)
153 ifcl 4080 . . . . . . . . . . . . . . . . 17 ((∅ ∈ 𝒫 𝑉 ∧ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ∈ 𝒫 𝑉)
154129, 152, 153sylancr 694 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ∈ 𝒫 𝑉)
155154elpwid 4118 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ⊆ 𝑉)
156155sseld 3567 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) → 𝑦𝑉))
157156pm4.71rd 665 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ↔ (𝑦𝑉𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
158128, 157bitr4d 270 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → ((𝑦𝑉 ∧ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑦)) ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
159120, 158syl5bb 271 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑦 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
160159eqrdv 2608 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
161160exp32 629 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (#‘𝐹) → ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
162161a2d 29 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
163116, 162syld 46 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
164163expcom 450 . . . . . 6 (𝑛 ∈ ℕ0 → (𝜑 → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))))
165164a2d 29 . . . . 5 (𝑛 ∈ ℕ0 → ((𝜑 → (𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑛))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝜑 → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑛 + 1)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))))
16642, 58, 74, 90, 106, 165nn0ind 11348 . . . 4 ((#‘𝐹) ∈ ℕ0 → (𝜑 → ((#‘𝐹) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))))
16721, 166mpcom 37 . . 3 (𝜑 → ((#‘𝐹) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})))
16820, 167mpd 15 . 2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(#‘𝐹)))))‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))
16916, 168eqtr3d 2646 1 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  #chash 12979   ∥ cdvds 14821   UMGrph cumg 25841   VDeg cvdg 26420   EulPaths ceup 26489 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-umgra 25842  df-vdgr 26421  df-eupa 26490 This theorem is referenced by:  eupath  26508
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