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Theorem inf3lem1 9564
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9571 for detailed description. (Contributed by NM, 28-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
inf3lem.2 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3 𝐴 ∈ V
inf3lem.4 𝐵 ∈ V
Assertion
Ref Expression
inf3lem1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))
Distinct variable group:   𝑥,𝑦,𝑤
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑤)

Proof of Theorem inf3lem1
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6842 . . 3 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
2 suceq 6383 . . . 4 (𝑣 = ∅ → suc 𝑣 = suc ∅)
32fveq2d 6846 . . 3 (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
41, 3sseq12d 3977 . 2 (𝑣 = ∅ → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5 fveq2 6842 . . 3 (𝑣 = 𝑢 → (𝐹𝑣) = (𝐹𝑢))
6 suceq 6383 . . . 4 (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
76fveq2d 6846 . . 3 (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
85, 7sseq12d 3977 . 2 (𝑣 = 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)))
9 fveq2 6842 . . 3 (𝑣 = suc 𝑢 → (𝐹𝑣) = (𝐹‘suc 𝑢))
10 suceq 6383 . . . 4 (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
1110fveq2d 6846 . . 3 (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
129, 11sseq12d 3977 . 2 (𝑣 = suc 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13 fveq2 6842 . . 3 (𝑣 = 𝐴 → (𝐹𝑣) = (𝐹𝐴))
14 suceq 6383 . . . 4 (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
1514fveq2d 6846 . . 3 (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
1613, 15sseq12d 3977 . 2 (𝑣 = 𝐴 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹𝐴) ⊆ (𝐹‘suc 𝐴)))
17 inf3lem.1 . . . 4 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
18 inf3lem.2 . . . 4 𝐹 = (rec(𝐺, ∅) ↾ ω)
19 inf3lem.3 . . . 4 𝐴 ∈ V
2017, 18, 19, 19inf3lemb 9561 . . 3 (𝐹‘∅) = ∅
21 0ss 4356 . . 3 ∅ ⊆ (𝐹‘suc ∅)
2220, 21eqsstri 3978 . 2 (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23 sstr2 3951 . . . . . . . 8 ((𝑣𝑥) ⊆ (𝐹𝑢) → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))
2423com12 32 . . . . . . 7 ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣𝑥) ⊆ (𝐹𝑢) → (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))
2524anim2d 612 . . . . . 6 ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
26 vex 3449 . . . . . . . . . 10 𝑢 ∈ V
2717, 18, 26, 19inf3lemc 9562 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹𝑢)))
2827eleq2d 2823 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹𝑢))))
29 vex 3449 . . . . . . . . 9 𝑣 ∈ V
30 fvex 6855 . . . . . . . . 9 (𝐹𝑢) ∈ V
3117, 18, 29, 30inf3lema 9560 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)))
3228, 31bitrdi 286 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢))))
33 peano2b 7819 . . . . . . . . . 10 (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
3426sucex 7741 . . . . . . . . . . 11 suc 𝑢 ∈ V
3517, 18, 34, 19inf3lemc 9562 . . . . . . . . . 10 (suc 𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
3633, 35sylbi 216 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
3736eleq2d 2823 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38 fvex 6855 . . . . . . . . 9 (𝐹‘suc 𝑢) ∈ V
3917, 18, 29, 38inf3lema 9560 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))
4037, 39bitrdi 286 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
4132, 40imbi12d 344 . . . . . 6 (𝑢 ∈ ω → ((𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)) ↔ ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))))
4225, 41syl5ibr 245 . . . . 5 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
4342imp 407 . . . 4 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
4443ssrdv 3950 . . 3 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
4544ex 413 . 2 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
464, 8, 12, 16, 22, 45finds 7835 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  cin 3909  wss 3910  c0 4282  cmpt 5188  cres 5635  suc csuc 6319  cfv 6496  ωcom 7802  reccrdg 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356
This theorem is referenced by:  inf3lem4  9567
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