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Theorem axdc3lem3 10446
Description: Simple substitution lemma for axdc3 10448. (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 2825 . 2 (𝐵𝑆𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
3 axdc3lem3.3 . . 3 𝐵 ∈ V
4 feq1 6698 . . . . 5 (𝑠 = 𝐵 → (𝑠:suc 𝑛𝐴𝐵:suc 𝑛𝐴))
5 fveq1 6890 . . . . . 6 (𝑠 = 𝐵 → (𝑠‘∅) = (𝐵‘∅))
65eqeq1d 2734 . . . . 5 (𝑠 = 𝐵 → ((𝑠‘∅) = 𝐶 ↔ (𝐵‘∅) = 𝐶))
7 fveq1 6890 . . . . . . 7 (𝑠 = 𝐵 → (𝑠‘suc 𝑘) = (𝐵‘suc 𝑘))
8 fveq1 6890 . . . . . . . 8 (𝑠 = 𝐵 → (𝑠𝑘) = (𝐵𝑘))
98fveq2d 6895 . . . . . . 7 (𝑠 = 𝐵 → (𝐹‘(𝑠𝑘)) = (𝐹‘(𝐵𝑘)))
107, 9eleq12d 2827 . . . . . 6 (𝑠 = 𝐵 → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1110ralbidv 3177 . . . . 5 (𝑠 = 𝐵 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
124, 6, 113anbi123d 1436 . . . 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 3178 . . 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 3668 . 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 6430 . . . . 5 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
1615feq2d 6703 . . . 4 (𝑛 = 𝑚 → (𝐵:suc 𝑛𝐴𝐵:suc 𝑚𝐴))
17 raleq 3322 . . . 4 (𝑛 = 𝑚 → (∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)) ↔ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1816, 173anbi13d 1438 . . 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 3235 . 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 296 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 1087   = wceq 1541  wcel 2106  {cab 2709  wral 3061  wrex 3070  Vcvv 3474  c0 4322  suc csuc 6366  wf 6539  cfv 6543  ωcom 7854
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  axdc3lem4  10447
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