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Theorem inf3lem1 9572
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9579 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 6846 . . 3 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
2 suceq 6387 . . . 4 (𝑣 = ∅ → suc 𝑣 = suc ∅)
32fveq2d 6850 . . 3 (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
41, 3sseq12d 3981 . 2 (𝑣 = ∅ → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5 fveq2 6846 . . 3 (𝑣 = 𝑢 → (𝐹𝑣) = (𝐹𝑢))
6 suceq 6387 . . . 4 (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
76fveq2d 6850 . . 3 (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
85, 7sseq12d 3981 . 2 (𝑣 = 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)))
9 fveq2 6846 . . 3 (𝑣 = suc 𝑢 → (𝐹𝑣) = (𝐹‘suc 𝑢))
10 suceq 6387 . . . 4 (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
1110fveq2d 6850 . . 3 (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
129, 11sseq12d 3981 . 2 (𝑣 = suc 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13 fveq2 6846 . . 3 (𝑣 = 𝐴 → (𝐹𝑣) = (𝐹𝐴))
14 suceq 6387 . . . 4 (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
1514fveq2d 6850 . . 3 (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
1613, 15sseq12d 3981 . 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 9569 . . 3 (𝐹‘∅) = ∅
21 0ss 4360 . . 3 ∅ ⊆ (𝐹‘suc ∅)
2220, 21eqsstri 3982 . 2 (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23 sstr2 3955 . . . . . . . 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 613 . . . . . 6 ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
26 vex 3451 . . . . . . . . . 10 𝑢 ∈ V
2717, 18, 26, 19inf3lemc 9570 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹𝑢)))
2827eleq2d 2820 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹𝑢))))
29 vex 3451 . . . . . . . . 9 𝑣 ∈ V
30 fvex 6859 . . . . . . . . 9 (𝐹𝑢) ∈ V
3117, 18, 29, 30inf3lema 9568 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)))
3228, 31bitrdi 287 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢))))
33 peano2b 7823 . . . . . . . . . 10 (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
3426sucex 7745 . . . . . . . . . . 11 suc 𝑢 ∈ V
3517, 18, 34, 19inf3lemc 9570 . . . . . . . . . 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 2820 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38 fvex 6859 . . . . . . . . 9 (𝐹‘suc 𝑢) ∈ V
3917, 18, 29, 38inf3lema 9568 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))
4037, 39bitrdi 287 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
4132, 40imbi12d 345 . . . . . 6 (𝑢 ∈ ω → ((𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)) ↔ ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))))
4225, 41imbitrrid 245 . . . . 5 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
4342imp 408 . . . 4 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
4443ssrdv 3954 . . 3 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
4544ex 414 . 2 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
464, 8, 12, 16, 22, 45finds 7839 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3447  cin 3913  wss 3914  c0 4286  cmpt 5192  cres 5639  suc csuc 6323  cfv 6500  ωcom 7806  reccrdg 8359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360
This theorem is referenced by:  inf3lem4  9575
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