Theorem acneq 9965

Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
acneq	⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)

Proof of Theorem acneq

Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2824	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2		oveq2 7375	. . . . 5 ⊢ (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶))
3		raleq 3292	. . . . . 6 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
4	3	exbidv 1923	. . . . 5 ⊢ (𝐴 = 𝐶 → (∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
5	2, 4	raleqbidv 3311	. . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
6	1, 5	anbi12d 633	. . 3 ⊢ (𝐴 = 𝐶 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)) ↔ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))))
7	6	abbidv 2802	. 2 ⊢ (𝐴 = 𝐶 → {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))})
8		df-acn 9866	. 2 ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
9		df-acn 9866	. 2 ⊢ AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
10	7, 8, 9	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051  Vcvv 3429   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  AC wacn 9862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-acn 9866

This theorem is referenced by:  acndom  9973  dfacacn  10064  dfac13  10065