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Theorem acneq 10083
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 2829 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2 oveq2 7439 . . . . 5 (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶))
3 raleq 3323 . . . . . 6 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
43exbidv 1921 . . . . 5 (𝐴 = 𝐶 → (∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
52, 4raleqbidv 3346 . . . 4 (𝐴 = 𝐶 → (∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
61, 5anbi12d 632 . . 3 (𝐴 = 𝐶 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) ↔ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))))
76abbidv 2808 . 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 9982 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
9 df-acn 9982 . 2 AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))}
107, 8, 93eqtr4g 2802 1 (𝐴 = 𝐶AC 𝐴 = AC 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  cfv 6561  (class class class)co 7431  m cmap 8866  AC wacn 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-acn 9982
This theorem is referenced by:  acndom  10091  dfacacn  10182  dfac13  10183
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