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Theorem axdc3lem3 9157
 Description: Simple substitution lemma for axdc3 9159. (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 2680 . 2 (𝐵𝑆𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
3 axdc3lem3.3 . . 3 𝐵 ∈ V
4 feq1 5939 . . . . 5 (𝑠 = 𝐵 → (𝑠:suc 𝑛𝐴𝐵:suc 𝑛𝐴))
5 fveq1 6102 . . . . . 6 (𝑠 = 𝐵 → (𝑠‘∅) = (𝐵‘∅))
65eqeq1d 2612 . . . . 5 (𝑠 = 𝐵 → ((𝑠‘∅) = 𝐶 ↔ (𝐵‘∅) = 𝐶))
7 fveq1 6102 . . . . . . 7 (𝑠 = 𝐵 → (𝑠‘suc 𝑘) = (𝐵‘suc 𝑘))
8 fveq1 6102 . . . . . . . 8 (𝑠 = 𝐵 → (𝑠𝑘) = (𝐵𝑘))
98fveq2d 6107 . . . . . . 7 (𝑠 = 𝐵 → (𝐹‘(𝑠𝑘)) = (𝐹‘(𝐵𝑘)))
107, 9eleq12d 2682 . . . . . 6 (𝑠 = 𝐵 → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1110ralbidv 2969 . . . . 5 (𝑠 = 𝐵 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
124, 6, 113anbi123d 1391 . . . 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 3034 . . 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 3319 . 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 5707 . . . . 5 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
1615feq2d 5944 . . . 4 (𝑛 = 𝑚 → (𝐵:suc 𝑛𝐴𝐵:suc 𝑚𝐴))
17 raleq 3115 . . . 4 (𝑛 = 𝑚 → (∀𝑘𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘)) ↔ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
1816, 173anbi13d 1393 . . 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 3148 . 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 285 1 (𝐵𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  suc csuc 5642  ⟶wf 5800  ‘cfv 5804  ωcom 6957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  axdc3lem4  9158
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