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Theorem bnj517 30209
 Description: Technical lemma for bnj518 30210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj517.1 (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj517.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj517 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
Distinct variable groups:   𝑖,𝑛,𝑦,𝐴   𝑖,𝐹,𝑛   𝑖,𝑁,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑖,𝑛)   𝜓(𝑦,𝑖,𝑛)   𝑅(𝑦,𝑖,𝑛)   𝐹(𝑦)   𝑁(𝑦)   𝑋(𝑦,𝑖,𝑛)

Proof of Theorem bnj517
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . . 6 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
2 simpl2 1058 . . . . . . 7 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝜑)
3 bnj517.1 . . . . . . 7 (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
42, 3sylib 207 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
51, 4sylan9eqr 2666 . . . . 5 ((((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) ∧ 𝑚 = ∅) → (𝐹𝑚) = pred(𝑋, 𝐴, 𝑅))
6 bnj213 30206 . . . . 5 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
75, 6syl6eqss 3618 . . . 4 ((((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) ∧ 𝑚 = ∅) → (𝐹𝑚) ⊆ 𝐴)
8 bnj517.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
9 r19.29r 3055 . . . . . . . . . 10 ((∃𝑖 ∈ ω 𝑚 = suc 𝑖 ∧ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → ∃𝑖 ∈ ω (𝑚 = suc 𝑖 ∧ (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))))
10 eleq1 2676 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → (𝑚𝑁 ↔ suc 𝑖𝑁))
1110biimpd 218 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → (𝑚𝑁 → suc 𝑖𝑁))
12 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑚 = suc 𝑖 → (𝐹𝑚) = (𝐹‘suc 𝑖))
1312eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
14 bnj213 30206 . . . . . . . . . . . . . . . . 17 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1514rgenw 2908 . . . . . . . . . . . . . . . 16 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
16 iunss 4497 . . . . . . . . . . . . . . . 16 ( 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
1715, 16mpbir 220 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18 sseq1 3589 . . . . . . . . . . . . . . 15 ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → ((𝐹𝑚) ⊆ 𝐴 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴))
1917, 18mpbiri 247 . . . . . . . . . . . . . 14 ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹𝑚) ⊆ 𝐴)
2013, 19syl6bir 243 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → ((𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹𝑚) ⊆ 𝐴))
2111, 20imim12d 79 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → ((suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴)))
2221imp 444 . . . . . . . . . . 11 ((𝑚 = suc 𝑖 ∧ (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴))
2322rexlimivw 3011 . . . . . . . . . 10 (∃𝑖 ∈ ω (𝑚 = suc 𝑖 ∧ (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴))
249, 23syl 17 . . . . . . . . 9 ((∃𝑖 ∈ ω 𝑚 = suc 𝑖 ∧ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴))
2524ex 449 . . . . . . . 8 (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴)))
2625com3l 87 . . . . . . 7 (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
278, 26sylbi 206 . . . . . 6 (𝜓 → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
28273ad2ant3 1077 . . . . 5 ((𝑁 ∈ ω ∧ 𝜑𝜓) → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
2928imp31 447 . . . 4 ((((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) ∧ ∃𝑖 ∈ ω 𝑚 = suc 𝑖) → (𝐹𝑚) ⊆ 𝐴)
30 simpr 476 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑚𝑁)
31 simpl1 1057 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑁 ∈ ω)
32 elnn 6967 . . . . . 6 ((𝑚𝑁𝑁 ∈ ω) → 𝑚 ∈ ω)
3330, 31, 32syl2anc 691 . . . . 5 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑚 ∈ ω)
34 nn0suc 6982 . . . . 5 (𝑚 ∈ ω → (𝑚 = ∅ ∨ ∃𝑖 ∈ ω 𝑚 = suc 𝑖))
3533, 34syl 17 . . . 4 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝑚 = ∅ ∨ ∃𝑖 ∈ ω 𝑚 = suc 𝑖))
367, 29, 35mpjaodan 823 . . 3 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝐹𝑚) ⊆ 𝐴)
3736ralrimiva 2949 . 2 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑚𝑁 (𝐹𝑚) ⊆ 𝐴)
38 fveq2 6103 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3938sseq1d 3595 . . 3 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ 𝐴 ↔ (𝐹𝑛) ⊆ 𝐴))
4039cbvralv 3147 . 2 (∀𝑚𝑁 (𝐹𝑚) ⊆ 𝐴 ↔ ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
4137, 40sylib 207 1 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  suc csuc 5642  ‘cfv 5804  ωcom 6957   predc-bnj14 30007 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-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fv 5812  df-om 6958  df-bnj14 30008 This theorem is referenced by:  bnj518  30210
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