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Theorem bnj517 34877
Description: Technical lemma for bnj518 34878. 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 6906 . . . . . 6 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
2 simpl2 1191 . . . . . . 7 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝜑)
3 bnj517.1 . . . . . . 7 (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
42, 3sylib 218 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
51, 4sylan9eqr 2796 . . . . 5 ((((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) ∧ 𝑚 = ∅) → (𝐹𝑚) = pred(𝑋, 𝐴, 𝑅))
6 bnj213 34874 . . . . 5 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
75, 6eqsstrdi 4049 . . . 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 3113 . . . . . . . . . 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 2826 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → (𝑚𝑁 ↔ suc 𝑖𝑁))
1110biimpd 229 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → (𝑚𝑁 → suc 𝑖𝑁))
12 fveqeq2 6915 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
13 bnj213 34874 . . . . . . . . . . . . . . . . 17 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1413rgenw 3062 . . . . . . . . . . . . . . . 16 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
15 iunss 5049 . . . . . . . . . . . . . . . 16 ( 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
1614, 15mpbir 231 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
17 sseq1 4020 . . . . . . . . . . . . . . 15 ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → ((𝐹𝑚) ⊆ 𝐴 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴))
1816, 17mpbiri 258 . . . . . . . . . . . . . 14 ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹𝑚) ⊆ 𝐴)
1912, 18biimtrrdi 254 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → ((𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹𝑚) ⊆ 𝐴))
2011, 19imim12d 81 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → ((suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴)))
2120imp 406 . . . . . . . . . . 11 ((𝑚 = suc 𝑖 ∧ (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴))
2221rexlimivw 3148 . . . . . . . . . 10 (∃𝑖 ∈ ω (𝑚 = suc 𝑖 ∧ (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴))
239, 22syl 17 . . . . . . . . 9 ((∃𝑖 ∈ ω 𝑚 = suc 𝑖 ∧ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴))
2423ex 412 . . . . . . . 8 (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (𝐹𝑚) ⊆ 𝐴)))
2524com3l 89 . . . . . . 7 (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
268, 25sylbi 217 . . . . . 6 (𝜓 → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
27263ad2ant3 1134 . . . . 5 ((𝑁 ∈ ω ∧ 𝜑𝜓) → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
2827imp31 417 . . . 4 ((((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) ∧ ∃𝑖 ∈ ω 𝑚 = suc 𝑖) → (𝐹𝑚) ⊆ 𝐴)
29 simpr 484 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑚𝑁)
30 simpl1 1190 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑁 ∈ ω)
31 elnn 7897 . . . . . 6 ((𝑚𝑁𝑁 ∈ ω) → 𝑚 ∈ ω)
3229, 30, 31syl2anc 584 . . . . 5 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑚 ∈ ω)
33 nn0suc 7916 . . . . 5 (𝑚 ∈ ω → (𝑚 = ∅ ∨ ∃𝑖 ∈ ω 𝑚 = suc 𝑖))
3432, 33syl 17 . . . 4 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝑚 = ∅ ∨ ∃𝑖 ∈ ω 𝑚 = suc 𝑖))
357, 28, 34mpjaodan 960 . . 3 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝐹𝑚) ⊆ 𝐴)
3635ralrimiva 3143 . 2 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑚𝑁 (𝐹𝑚) ⊆ 𝐴)
37 fveq2 6906 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3837sseq1d 4026 . . 3 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ 𝐴 ↔ (𝐹𝑛) ⊆ 𝐴))
3938cbvralvw 3234 . 2 (∀𝑚𝑁 (𝐹𝑚) ⊆ 𝐴 ↔ ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
4036, 39sylib 218 1 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  wss 3962  c0 4338   ciun 4995  suc csuc 6387  cfv 6562  ωcom 7886   predc-bnj14 34680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fv 6570  df-om 7887  df-bnj14 34681
This theorem is referenced by:  bnj518  34878
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