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Mirrors > Home > MPE Home > Th. List > acneq | Structured version Visualization version GIF version |
Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acneq | ⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2898 | . . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V)) | |
2 | oveq2 7156 | . . . . 5 ⊢ (𝐴 = 𝐶 → ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)) | |
3 | raleq 3404 | . . . . . 6 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))) | |
4 | 3 | exbidv 1916 | . . . . 5 ⊢ (𝐴 = 𝐶 → (∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))) |
5 | 2, 4 | raleqbidv 3400 | . . . 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 632 | . . 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 2883 | . 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 9363 | . 2 ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} | |
9 | df-acn 9363 | . 2 ⊢ AC 𝐶 = {𝑥 ∣ (𝐶 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐶)∃𝑔∀𝑦 ∈ 𝐶 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} | |
10 | 7, 8, 9 | 3eqtr4g 2879 | 1 ⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∃wex 1774 ∈ wcel 2108 {cab 2797 ∀wral 3136 Vcvv 3493 ∖ cdif 3931 ∅c0 4289 𝒫 cpw 4537 {csn 4559 ‘cfv 6348 (class class class)co 7148 ↑m cmap 8398 AC wacn 9359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7151 df-acn 9363 |
This theorem is referenced by: acndom 9469 dfacacn 9559 dfac13 9560 |
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