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Theorem acneq 9799
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 2826 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2 oveq2 7283 . . . . 5 (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶))
3 raleq 3342 . . . . . 6 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
43exbidv 1924 . . . . 5 (𝐴 = 𝐶 → (∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
52, 4raleqbidv 3336 . . . 4 (𝐴 = 𝐶 → (∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
61, 5anbi12d 631 . . 3 (𝐴 = 𝐶 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) ↔ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))))
76abbidv 2807 . 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 9700 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
9 df-acn 9700 . 2 AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))}
107, 8, 93eqtr4g 2803 1 (𝐴 = 𝐶AC 𝐴 = AC 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wral 3064  Vcvv 3432  cdif 3884  c0 4256  𝒫 cpw 4533  {csn 4561  cfv 6433  (class class class)co 7275  m cmap 8615  AC wacn 9696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-acn 9700
This theorem is referenced by:  acndom  9807  dfacacn  9897  dfac13  9898
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