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Theorem inf3lema 8689
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8700 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
inf3lema (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
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 inf3lema
Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq1 3958 . . 3 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
21sseq1d 3781 . 2 (𝑓 = 𝐴 → ((𝑓𝑥) ⊆ 𝐵 ↔ (𝐴𝑥) ⊆ 𝐵))
3 inf3lem.4 . . 3 𝐵 ∈ V
4 sseq2 3776 . . . . 5 (𝑣 = 𝐵 → ((𝑓𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝐵))
54rabbidv 3339 . . . 4 (𝑣 = 𝐵 → {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
6 inf3lem.1 . . . . 5 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
7 sseq2 3776 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑤𝑥) ⊆ 𝑦 ↔ (𝑤𝑥) ⊆ 𝑣))
87rabbidv 3339 . . . . . . 7 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣})
9 ineq1 3958 . . . . . . . . 9 (𝑤 = 𝑓 → (𝑤𝑥) = (𝑓𝑥))
109sseq1d 3781 . . . . . . . 8 (𝑤 = 𝑓 → ((𝑤𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝑣))
1110cbvrabv 3349 . . . . . . 7 {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣}
128, 11syl6eq 2821 . . . . . 6 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
1312cbvmptv 4885 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
146, 13eqtri 2793 . . . 4 𝐺 = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
15 vex 3354 . . . . 5 𝑥 ∈ V
1615rabex 4947 . . . 4 {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵} ∈ V
175, 14, 16fvmpt 6426 . . 3 (𝐵 ∈ V → (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
183, 17ax-mp 5 . 2 (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵}
192, 18elrab2 3518 1 (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  cin 3722  wss 3723  c0 4063  cmpt 4864  cres 5252  cfv 6030  ωcom 7216  reccrdg 7662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038
This theorem is referenced by:  inf3lemd  8692  inf3lem1  8693  inf3lem2  8694  inf3lem3  8695
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