Theorem acneq 7552

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 𝑧 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2301	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
2		oveq2 6087	. . . . 5 ⊢ (𝐴 = 𝐶 → ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴) = ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐶))
3		raleq 2749	. . . . . 6 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
4	3	exbidv 1878	. . . . 5 ⊢ (𝐴 = 𝐶 → (∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
5	2, 4	raleqbidv 2765	. . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
6	1, 5	anbi12d 477	. . 3 ⊢ (𝐴 = 𝐶 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)) ↔ (𝐶 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))))
7	6	abbidv 2358	. 2 ⊢ (𝐴 = 𝐶 → {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))})
8		df-acnm 7519	. 2 ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
9		df-acnm 7519	. 2 ⊢ AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
10	7, 8, 9	3eqtr4g 2296	1 ⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  {crab 2532  Vcvv 2821  𝒫 cpw 3688  ‘cfv 5375  (class class class)co 6079   ↑_𝑚 cmap 6916  AC wacn 7517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6082  df-acnm 7519

This theorem is referenced by: (None)