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Theorem axdc3lem3 10389
Description: Simple substitution lemma for axdc3 10391. (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 2830 . 2 (𝐵𝑆𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
3 axdc3lem3.3 . . 3 𝐵 ∈ V
4 feq1 6650 . . . . 5 (𝑠 = 𝐵 → (𝑠:suc 𝑛𝐴𝐵:suc 𝑛𝐴))
5 fveq1 6842 . . . . . 6 (𝑠 = 𝐵 → (𝑠‘∅) = (𝐵‘∅))
65eqeq1d 2739 . . . . 5 (𝑠 = 𝐵 → ((𝑠‘∅) = 𝐶 ↔ (𝐵‘∅) = 𝐶))
7 fveq1 6842 . . . . . . 7 (𝑠 = 𝐵 → (𝑠‘suc 𝑘) = (𝐵‘suc 𝑘))
8 fveq1 6842 . . . . . . . 8 (𝑠 = 𝐵 → (𝑠𝑘) = (𝐵𝑘))
98fveq2d 6847 . . . . . . 7 (𝑠 = 𝐵 → (𝐹‘(𝑠𝑘)) = (𝐹‘(𝐵𝑘)))
107, 9eleq12d 2832 . . . . . 6 (𝑠 = 𝐵 → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1110ralbidv 3175 . . . . 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 3176 . . 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 3631 . 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 6384 . . . . 5 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
1615feq2d 6655 . . . 4 (𝑛 = 𝑚 → (𝐵:suc 𝑛𝐴𝐵:suc 𝑚𝐴))
17 raleq 3310 . . . 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 3227 . 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 2714  wral 3065  wrex 3074  Vcvv 3446  c0 4283  suc csuc 6320  wf 6493  cfv 6497  ωcom 7803
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  axdc3lem4  10390
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