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Theorem inf3lem1 9088
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9095 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 6661 . . 3 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
2 suceq 6243 . . . 4 (𝑣 = ∅ → suc 𝑣 = suc ∅)
32fveq2d 6665 . . 3 (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
41, 3sseq12d 3986 . 2 (𝑣 = ∅ → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5 fveq2 6661 . . 3 (𝑣 = 𝑢 → (𝐹𝑣) = (𝐹𝑢))
6 suceq 6243 . . . 4 (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
76fveq2d 6665 . . 3 (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
85, 7sseq12d 3986 . 2 (𝑣 = 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)))
9 fveq2 6661 . . 3 (𝑣 = suc 𝑢 → (𝐹𝑣) = (𝐹‘suc 𝑢))
10 suceq 6243 . . . 4 (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
1110fveq2d 6665 . . 3 (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
129, 11sseq12d 3986 . 2 (𝑣 = suc 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13 fveq2 6661 . . 3 (𝑣 = 𝐴 → (𝐹𝑣) = (𝐹𝐴))
14 suceq 6243 . . . 4 (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
1514fveq2d 6665 . . 3 (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
1613, 15sseq12d 3986 . 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 9085 . . 3 (𝐹‘∅) = ∅
21 0ss 4333 . . 3 ∅ ⊆ (𝐹‘suc ∅)
2220, 21eqsstri 3987 . 2 (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23 sstr2 3960 . . . . . . . 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 614 . . . . . 6 ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
26 vex 3483 . . . . . . . . . 10 𝑢 ∈ V
2717, 18, 26, 19inf3lemc 9086 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹𝑢)))
2827eleq2d 2901 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹𝑢))))
29 vex 3483 . . . . . . . . 9 𝑣 ∈ V
30 fvex 6674 . . . . . . . . 9 (𝐹𝑢) ∈ V
3117, 18, 29, 30inf3lema 9084 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)))
3228, 31syl6bb 290 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢))))
33 peano2b 7590 . . . . . . . . . 10 (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
3426sucex 7520 . . . . . . . . . . 11 suc 𝑢 ∈ V
3517, 18, 34, 19inf3lemc 9086 . . . . . . . . . 10 (suc 𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
3633, 35sylbi 220 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
3736eleq2d 2901 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38 fvex 6674 . . . . . . . . 9 (𝐹‘suc 𝑢) ∈ V
3917, 18, 29, 38inf3lema 9084 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))
4037, 39syl6bb 290 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
4132, 40imbi12d 348 . . . . . 6 (𝑢 ∈ ω → ((𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)) ↔ ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))))
4225, 41syl5ibr 249 . . . . 5 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
4342imp 410 . . . 4 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
4443ssrdv 3959 . . 3 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
4544ex 416 . 2 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
464, 8, 12, 16, 22, 45finds 7603 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  cin 3918  wss 3919  c0 4276  cmpt 5132  cres 5544  suc csuc 6180  cfv 6343  ωcom 7574  reccrdg 8041
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042
This theorem is referenced by:  inf3lem4  9091
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