Theorem acneq 10023

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 2851	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2		oveq2 7418	. . . . 5 ⊢ (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶))
3		raleq 3320	. . . . . 6 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
4	3	exbidv 1951	. . . . 5 ⊢ (𝐴 = 𝐶 → (∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
5	2, 4	raleqbidv 3338	. . . 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 643	. . 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 2829	. 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 9924	. 2 ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
9		df-acn 9924	. 2 ⊢ AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
10	7, 8, 9	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  Vcvv 3455   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  AC wacn 9920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-acn 9924

This theorem is referenced by:  acndom  10031  dfacacn  10121  dfac13  10122