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Theorem axdc3lem3 9312
Description: Simple substitution lemma for axdc3 9314. (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 2722 . 2 (𝐵𝑆𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
3 axdc3lem3.3 . . 3 𝐵 ∈ V
4 feq1 6064 . . . . 5 (𝑠 = 𝐵 → (𝑠:suc 𝑛𝐴𝐵:suc 𝑛𝐴))
5 fveq1 6228 . . . . . 6 (𝑠 = 𝐵 → (𝑠‘∅) = (𝐵‘∅))
65eqeq1d 2653 . . . . 5 (𝑠 = 𝐵 → ((𝑠‘∅) = 𝐶 ↔ (𝐵‘∅) = 𝐶))
7 fveq1 6228 . . . . . . 7 (𝑠 = 𝐵 → (𝑠‘suc 𝑘) = (𝐵‘suc 𝑘))
8 fveq1 6228 . . . . . . . 8 (𝑠 = 𝐵 → (𝑠𝑘) = (𝐵𝑘))
98fveq2d 6233 . . . . . . 7 (𝑠 = 𝐵 → (𝐹‘(𝑠𝑘)) = (𝐹‘(𝐵𝑘)))
107, 9eleq12d 2724 . . . . . 6 (𝑠 = 𝐵 → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1110ralbidv 3015 . . . . 5 (𝑠 = 𝐵 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
124, 6, 113anbi123d 1439 . . . 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 3081 . . 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 3382 . 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 5828 . . . . 5 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
1615feq2d 6069 . . . 4 (𝑛 = 𝑚 → (𝐵:suc 𝑛𝐴𝐵:suc 𝑚𝐴))
17 raleq 3168 . . . 4 (𝑛 = 𝑚 → (∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)) ↔ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1816, 173anbi13d 1441 . . 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)))))
1918cbvrexv 3202 . 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 286 1 (𝐵𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  c0 3948  suc csuc 5763  wf 5922  cfv 5926  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  axdc3lem4  9313
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