MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inf3lem1 Structured version   Visualization version   GIF version

Theorem inf3lem1 8740
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8747 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 6375 . . 3 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
2 suceq 5973 . . . 4 (𝑣 = ∅ → suc 𝑣 = suc ∅)
32fveq2d 6379 . . 3 (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
41, 3sseq12d 3794 . 2 (𝑣 = ∅ → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5 fveq2 6375 . . 3 (𝑣 = 𝑢 → (𝐹𝑣) = (𝐹𝑢))
6 suceq 5973 . . . 4 (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
76fveq2d 6379 . . 3 (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
85, 7sseq12d 3794 . 2 (𝑣 = 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)))
9 fveq2 6375 . . 3 (𝑣 = suc 𝑢 → (𝐹𝑣) = (𝐹‘suc 𝑢))
10 suceq 5973 . . . 4 (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
1110fveq2d 6379 . . 3 (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
129, 11sseq12d 3794 . 2 (𝑣 = suc 𝑢 → ((𝐹𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13 fveq2 6375 . . 3 (𝑣 = 𝐴 → (𝐹𝑣) = (𝐹𝐴))
14 suceq 5973 . . . 4 (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
1514fveq2d 6379 . . 3 (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
1613, 15sseq12d 3794 . 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 8737 . . 3 (𝐹‘∅) = ∅
21 0ss 4134 . . 3 ∅ ⊆ (𝐹‘suc ∅)
2220, 21eqsstri 3795 . 2 (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23 sstr2 3768 . . . . . . . 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 605 . . . . . 6 ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
26 vex 3353 . . . . . . . . . 10 𝑢 ∈ V
2717, 18, 26, 19inf3lemc 8738 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹𝑢)))
2827eleq2d 2830 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹𝑢))))
29 vex 3353 . . . . . . . . 9 𝑣 ∈ V
30 fvex 6388 . . . . . . . . 9 (𝐹𝑢) ∈ V
3117, 18, 29, 30inf3lema 8736 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)))
3228, 31syl6bb 278 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢))))
33 peano2b 7279 . . . . . . . . . 10 (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
3426sucex 7209 . . . . . . . . . . 11 suc 𝑢 ∈ V
3517, 18, 34, 19inf3lemc 8738 . . . . . . . . . 10 (suc 𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
3633, 35sylbi 208 . . . . . . . . 9 (𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
3736eleq2d 2830 . . . . . . . 8 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38 fvex 6388 . . . . . . . . 9 (𝐹‘suc 𝑢) ∈ V
3917, 18, 29, 38inf3lema 8736 . . . . . . . 8 (𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢)) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))
4037, 39syl6bb 278 . . . . . . 7 (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢))))
4132, 40imbi12d 335 . . . . . 6 (𝑢 ∈ ω → ((𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)) ↔ ((𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹𝑢)) → (𝑣𝑥 ∧ (𝑣𝑥) ⊆ (𝐹‘suc 𝑢)))))
4225, 41syl5ibr 237 . . . . 5 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
4342imp 395 . . . 4 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
4443ssrdv 3767 . . 3 ((𝑢 ∈ ω ∧ (𝐹𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
4544ex 401 . 2 (𝑢 ∈ ω → ((𝐹𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
464, 8, 12, 16, 22, 45finds 7290 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  cin 3731  wss 3732  c0 4079  cmpt 4888  cres 5279  suc csuc 5910  cfv 6068  ωcom 7263  reccrdg 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710
This theorem is referenced by:  inf3lem4  8743
  Copyright terms: Public domain W3C validator