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Theorem acneq 10081
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 2827 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2 oveq2 7439 . . . . 5 (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶))
3 raleq 3321 . . . . . 6 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
43exbidv 1919 . . . . 5 (𝐴 = 𝐶 → (∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦) ↔ ∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦)))
52, 4raleqbidv 3344 . . . 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 2806 . 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 9980 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
9 df-acn 9980 . 2 AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔𝑦𝐶 (𝑔𝑦) ∈ (𝑓𝑦))}
107, 8, 93eqtr4g 2800 1 (𝐴 = 𝐶AC 𝐴 = AC 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  cdif 3960  c0 4339  𝒫 cpw 4605  {csn 4631  cfv 6563  (class class class)co 7431  m cmap 8865  AC wacn 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-acn 9980
This theorem is referenced by:  acndom  10089  dfacacn  10180  dfac13  10181
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