|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > isacn | Structured version Visualization version GIF version | ||
| Description: The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| isacn | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pweq 4613 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋) | |
| 2 | 1 | difeq1d 4124 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝒫 𝑦 ∖ {∅}) = (𝒫 𝑋 ∖ {∅})) | 
| 3 | 2 | oveq1d 7447 | . . . . 5 ⊢ (𝑦 = 𝑋 → ((𝒫 𝑦 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) | 
| 4 | 3 | raleqdv 3325 | . . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | 
| 5 | 4 | anbi2d 630 | . . 3 ⊢ (𝑦 = 𝑋 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))) | 
| 6 | df-acn 9983 | . . 3 ⊢ AC 𝐴 = {𝑦 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))} | |
| 7 | 5, 6 | elab2g 3679 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ AC 𝐴 ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))) | 
| 8 | elex 3500 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
| 9 | biid 261 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | |
| 10 | 9 | baib 535 | . . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | 
| 11 | 8, 10 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | 
| 12 | 7, 11 | sylan9bb 509 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∖ cdif 3947 ∅c0 4332 𝒫 cpw 4599 {csn 4625 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 AC wacn 9979 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-acn 9983 | 
| This theorem is referenced by: acni 10086 numacn 10090 finacn 10091 acndom 10092 acndom2 10095 acncc 10481 | 
| Copyright terms: Public domain | W3C validator |