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Theorem axdc3lem3 10447
Description: Simple substitution lemma for axdc3 10449. (Contributed by Mario Carneiro, 27-Jan-2013.)
Hypotheses
Ref Expression
axdc3lem3.1 𝐴 ∈ V
axdc3lem3.2 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
axdc3lem3.3 𝐵 ∈ V
Assertion
Ref Expression
axdc3lem3 (𝐵𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
Distinct variable groups:   𝐴,𝑚,𝑛   𝐴,𝑠,𝑛   𝐵,𝑘,𝑚,𝑛   𝐵,𝑠,𝑘   𝐶,𝑚,𝑛   𝐶,𝑠   𝑚,𝐹,𝑛   𝐹,𝑠
Allowed substitution hints:   𝐴(𝑘)   𝐶(𝑘)   𝑆(𝑘,𝑚,𝑛,𝑠)   𝐹(𝑘)

Proof of Theorem axdc3lem3
StepHypRef Expression
1 axdc3lem3.2 . . 3 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
21eleq2i 2826 . 2 (𝐵𝑆𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
3 axdc3lem3.3 . . 3 𝐵 ∈ V
4 feq1 6699 . . . . 5 (𝑠 = 𝐵 → (𝑠:suc 𝑛𝐴𝐵:suc 𝑛𝐴))
5 fveq1 6891 . . . . . 6 (𝑠 = 𝐵 → (𝑠‘∅) = (𝐵‘∅))
65eqeq1d 2735 . . . . 5 (𝑠 = 𝐵 → ((𝑠‘∅) = 𝐶 ↔ (𝐵‘∅) = 𝐶))
7 fveq1 6891 . . . . . . 7 (𝑠 = 𝐵 → (𝑠‘suc 𝑘) = (𝐵‘suc 𝑘))
8 fveq1 6891 . . . . . . . 8 (𝑠 = 𝐵 → (𝑠𝑘) = (𝐵𝑘))
98fveq2d 6896 . . . . . . 7 (𝑠 = 𝐵 → (𝐹‘(𝑠𝑘)) = (𝐹‘(𝐵𝑘)))
107, 9eleq12d 2828 . . . . . 6 (𝑠 = 𝐵 → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1110ralbidv 3178 . . . . 5 (𝑠 = 𝐵 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
124, 6, 113anbi123d 1437 . . . 4 (𝑠 = 𝐵 → ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) ↔ (𝐵:suc 𝑛𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)))))
1312rexbidv 3179 . . 3 (𝑠 = 𝐵 → (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) ↔ ∃𝑛 ∈ ω (𝐵:suc 𝑛𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)))))
143, 13elab 3669 . 2 (𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ↔ ∃𝑛 ∈ ω (𝐵:suc 𝑛𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
15 suceq 6431 . . . . 5 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
1615feq2d 6704 . . . 4 (𝑛 = 𝑚 → (𝐵:suc 𝑛𝐴𝐵:suc 𝑚𝐴))
17 raleq 3323 . . . 4 (𝑛 = 𝑚 → (∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)) ↔ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1816, 173anbi13d 1439 . . 3 (𝑛 = 𝑚 → ((𝐵:suc 𝑛𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))) ↔ (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)))))
1918cbvrexvw 3236 . 2 (∃𝑛 ∈ ω (𝐵:suc 𝑛𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))) ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
202, 14, 193bitri 297 1 (𝐵𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  c0 4323  suc csuc 6367  wf 6540  cfv 6544  ωcom 7855
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  axdc3lem4  10448
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