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Theorem isf32lem2 9059
 Description: Lemma for isfin3-2 9072. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (𝜑𝐹:ω⟶𝒫 𝐺)
isf32lem.b (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
isf32lem.c (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem2 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
Distinct variable groups:   𝑥,𝑎   𝐺,𝑎   𝜑,𝑎,𝑥   𝐴,𝑎,𝑥   𝐹,𝑎,𝑥
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem isf32lem2
Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isf32lem.c . . . . 5 (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
21adantr 480 . . . 4 ((𝜑𝐴 ∈ ω) → ¬ ran 𝐹 ∈ ran 𝐹)
3 isf32lem.a . . . . . . . . . 10 (𝜑𝐹:ω⟶𝒫 𝐺)
4 ffn 5958 . . . . . . . . . 10 (𝐹:ω⟶𝒫 𝐺𝐹 Fn ω)
53, 4syl 17 . . . . . . . . 9 (𝜑𝐹 Fn ω)
6 peano2 6978 . . . . . . . . 9 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
7 fnfvelrn 6264 . . . . . . . . 9 ((𝐹 Fn ω ∧ suc 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
85, 6, 7syl2an 493 . . . . . . . 8 ((𝜑𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
98adantr 480 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
10 intss1 4427 . . . . . . 7 ((𝐹‘suc 𝐴) ∈ ran 𝐹 ran 𝐹 ⊆ (𝐹‘suc 𝐴))
119, 10syl 17 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 ⊆ (𝐹‘suc 𝐴))
12 fvelrnb 6153 . . . . . . . . . . 11 (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
135, 12syl 17 . . . . . . . . . 10 (𝜑 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
1413ad2antrr 758 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
15 simplrr 797 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → 𝑐 ∈ ω)
166ad3antlr 763 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → suc 𝐴 ∈ ω)
17 simpr 476 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → suc 𝐴𝑐)
18 simplrl 796 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
19 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝐴 → (𝐹𝑏) = (𝐹‘suc 𝐴))
2019eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝐴 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
2120imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑏 = suc 𝐴 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴))))
22 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
2322eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑑)))
2423imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑑 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑑))))
25 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝑑 → (𝐹𝑏) = (𝐹‘suc 𝑑))
2625eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
2726imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑏 = suc 𝑑 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
28 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
2928eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑐)))
3029imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑐))))
31 eqid 2610 . . . . . . . . . . . . . . . . . 18 (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)
32312a1i 12 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ ω → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
33 elex 3185 . . . . . . . . . . . . . . . . . . . . . . . 24 (suc 𝐴 ∈ ω → suc 𝐴 ∈ V)
34 sucexb 6901 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3533, 34sylibr 223 . . . . . . . . . . . . . . . . . . . . . . 23 (suc 𝐴 ∈ ω → 𝐴 ∈ V)
3635adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → 𝐴 ∈ V)
37 sucssel 5736 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ V → (suc 𝐴𝑑𝐴𝑑))
3836, 37syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴𝑑𝐴𝑑))
3938imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → 𝐴𝑑)
40 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → (𝐴𝑎𝐴𝑑))
41 suceq 5707 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑑 → suc 𝑎 = suc 𝑑)
4241fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑑))
43 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹𝑎) = (𝐹𝑑))
4442, 43eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → ((𝐹‘suc 𝑎) = (𝐹𝑎) ↔ (𝐹‘suc 𝑑) = (𝐹𝑑)))
4540, 44imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑑 → ((𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4645rspcv 3278 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ω → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4746com23 84 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ ω → (𝐴𝑑 → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4847ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (𝐴𝑑 → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4939, 48mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑)))
50 eqtr3 2631 . . . . . . . . . . . . . . . . . . . 20 (((𝐹‘suc 𝐴) = (𝐹𝑑) ∧ (𝐹‘suc 𝑑) = (𝐹𝑑)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))
5150expcom 450 . . . . . . . . . . . . . . . . . . 19 ((𝐹‘suc 𝑑) = (𝐹𝑑) → ((𝐹‘suc 𝐴) = (𝐹𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
5249, 51syl6 34 . . . . . . . . . . . . . . . . . 18 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ((𝐹‘suc 𝐴) = (𝐹𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
5352a2d 29 . . . . . . . . . . . . . . . . 17 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑑)) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
5421, 24, 27, 30, 32, 53findsg 6985 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑐) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑐)))
5554impr 647 . . . . . . . . . . . . . . 15 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ (suc 𝐴𝑐 ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))) → (𝐹‘suc 𝐴) = (𝐹𝑐))
5615, 16, 17, 18, 55syl22anc 1319 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) = (𝐹𝑐))
57 eqimss 3620 . . . . . . . . . . . . . 14 ((𝐹‘suc 𝐴) = (𝐹𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
5856, 57syl 17 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
596ad3antlr 763 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → suc 𝐴 ∈ ω)
60 simplrr 797 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ∈ ω)
61 simpr 476 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ⊆ suc 𝐴)
62 simplll 794 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝜑)
63 isf32lem.b . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
643, 63, 1isf32lem1 9058 . . . . . . . . . . . . . 14 (((suc 𝐴 ∈ ω ∧ 𝑐 ∈ ω) ∧ (𝑐 ⊆ suc 𝐴𝜑)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
6559, 60, 61, 62, 64syl22anc 1319 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
66 nnord 6965 . . . . . . . . . . . . . . . 16 (suc 𝐴 ∈ ω → Ord suc 𝐴)
676, 66syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → Ord suc 𝐴)
6867ad2antlr 759 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord suc 𝐴)
69 nnord 6965 . . . . . . . . . . . . . . 15 (𝑐 ∈ ω → Ord 𝑐)
7069ad2antll 761 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord 𝑐)
71 ordtri2or2 5740 . . . . . . . . . . . . . 14 ((Ord suc 𝐴 ∧ Ord 𝑐) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7268, 70, 71syl2anc 691 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7358, 65, 72mpjaodan 823 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
7473anassrs 678 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
75 sseq2 3590 . . . . . . . . . . 11 ((𝐹𝑐) = 𝑏 → ((𝐹‘suc 𝐴) ⊆ (𝐹𝑐) ↔ (𝐹‘suc 𝐴) ⊆ 𝑏))
7674, 75syl5ibcom 234 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → ((𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7776rexlimdva 3013 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (∃𝑐 ∈ ω (𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7814, 77sylbid 229 . . . . . . . 8 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7978ralrimiv 2948 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
80 ssint 4428 . . . . . . 7 ((𝐹‘suc 𝐴) ⊆ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
8179, 80sylibr 223 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ⊆ ran 𝐹)
8211, 81eqssd 3585 . . . . 5 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 = (𝐹‘suc 𝐴))
8382, 9eqeltrd 2688 . . . 4 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 ∈ ran 𝐹)
842, 83mtand 689 . . 3 ((𝜑𝐴 ∈ ω) → ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
85 rexnal 2978 . . 3 (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
8684, 85sylibr 223 . 2 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
87 suceq 5707 . . . . . . . 8 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
8887fveq2d 6107 . . . . . . 7 (𝑥 = 𝑎 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑎))
89 fveq2 6103 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
9088, 89sseq12d 3597 . . . . . 6 (𝑥 = 𝑎 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑎) ⊆ (𝐹𝑎)))
9190cbvralv 3147 . . . . 5 (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9263, 91sylib 207 . . . 4 (𝜑 → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9392adantr 480 . . 3 ((𝜑𝐴 ∈ ω) → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
94 pm4.61 441 . . . . 5 (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
95 dfpss2 3654 . . . . . . 7 ((𝐹‘suc 𝑎) ⊊ (𝐹𝑎) ↔ ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
9695simplbi2 653 . . . . . 6 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐹‘suc 𝑎) = (𝐹𝑎) → (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
9796anim2d 587 . . . . 5 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ((𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9894, 97syl5bi 231 . . . 4 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9998ralimi 2936 . . 3 (∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ∀𝑎 ∈ ω (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
100 rexim 2991 . . 3 (∀𝑎 ∈ ω (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))) → (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
10193, 99, 1003syl 18 . 2 ((𝜑𝐴 ∈ ω) → (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
10286, 101mpd 15 1 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541  𝒫 cpw 4108  ∩ cint 4410  ran crn 5039  Ord word 5639  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  ωcom 6957 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-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-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-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-om 6958 This theorem is referenced by:  isf32lem5  9062
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