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Theorem bnj517 31081
Description: Technical lemma for bnj518 31082. 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 6229 . . . . . 6 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
2 simpl2 1085 . . . . . . 7 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝜑)
3 bnj517.1 . . . . . . 7 (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
42, 3sylib 208 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
51, 4sylan9eqr 2707 . . . . 5 ((((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) ∧ 𝑚 = ∅) → (𝐹𝑚) = pred(𝑋, 𝐴, 𝑅))
6 bnj213 31078 . . . . 5 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
75, 6syl6eqss 3688 . . . 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 3102 . . . . . . . . . 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 2718 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → (𝑚𝑁 ↔ suc 𝑖𝑁))
1110biimpd 219 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → (𝑚𝑁 → suc 𝑖𝑁))
12 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑚 = suc 𝑖 → (𝐹𝑚) = (𝐹‘suc 𝑖))
1312eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
14 bnj213 31078 . . . . . . . . . . . . . . . . 17 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1514rgenw 2953 . . . . . . . . . . . . . . . 16 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
16 iunss 4593 . . . . . . . . . . . . . . . 16 ( 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
1715, 16mpbir 221 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18 sseq1 3659 . . . . . . . . . . . . . . 15 ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → ((𝐹𝑚) ⊆ 𝐴 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴))
1917, 18mpbiri 248 . . . . . . . . . . . . . 14 ((𝐹𝑚) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹𝑚) ⊆ 𝐴)
2013, 19syl6bir 244 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → ((𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹𝑚) ⊆ 𝐴))
2111, 20imim12d 81 . . . . . . . . . . . 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 3058 . . . . . . . . . 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 89 . . . . . . 7 (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
278, 26sylbi 207 . . . . . 6 (𝜓 → (𝑚𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹𝑚) ⊆ 𝐴)))
28273ad2ant3 1104 . . . . 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 1084 . . . . . 6 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑁 ∈ ω)
32 elnn 7117 . . . . . 6 ((𝑚𝑁𝑁 ∈ ω) → 𝑚 ∈ ω)
3330, 31, 32syl2anc 694 . . . . 5 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → 𝑚 ∈ ω)
34 nn0suc 7132 . . . . 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 844 . . 3 (((𝑁 ∈ ω ∧ 𝜑𝜓) ∧ 𝑚𝑁) → (𝐹𝑚) ⊆ 𝐴)
3736ralrimiva 2995 . 2 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑚𝑁 (𝐹𝑚) ⊆ 𝐴)
38 fveq2 6229 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3938sseq1d 3665 . . 3 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ 𝐴 ↔ (𝐹𝑛) ⊆ 𝐴))
4039cbvralv 3201 . 2 (∀𝑚𝑁 (𝐹𝑚) ⊆ 𝐴 ↔ ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
4137, 40sylib 208 1 ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  c0 3948   ciun 4552  suc csuc 5763  cfv 5926  ωcom 7107   predc-bnj14 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fv 5934  df-om 7108  df-bnj14 30883
This theorem is referenced by:  bnj518  31082
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