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Theorem acneq 8749
 Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acneq (𝐴 = 𝐶AC 𝐴 = AC 𝐶)

Proof of Theorem acneq
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2676 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2 oveq2 6557 . . . . 5 (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐶))
3 raleq 3115 . . . . . 6 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
43exbidv 1837 . . . . 5 (𝐴 = 𝐶 → (∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
52, 4raleqbidv 3129 . . . 4 (𝐴 = 𝐶 → (∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
61, 5anbi12d 743 . . 3 (𝐴 = 𝐶 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) ↔ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))))
76abbidv 2728 . 2 (𝐴 = 𝐶 → {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))} = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))})
8 df-acn 8651 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
9 df-acn 8651 . 2 AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))}
107, 8, 93eqtr4g 2669 1 (𝐴 = 𝐶AC 𝐴 = AC 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  AC wacn 8647 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-iota 5768  df-fv 5812  df-ov 6552  df-acn 8651 This theorem is referenced by:  acndom  8757  dfacacn  8846  dfac13  8847
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