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| Mirrors > Home > MPE Home > Th. List > xpcomf1o | Structured version Visualization version GIF version | ||
| Description: The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.) |
| Ref | Expression |
|---|---|
| xpcomf1o.1 | ⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) |
| Ref | Expression |
|---|---|
| xpcomf1o | ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relxp 5703 | . . . 4 ⊢ Rel (𝐴 × 𝐵) | |
| 2 | cnvf1o 8136 | . . . 4 ⊢ (Rel (𝐴 × 𝐵) → (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) |
| 4 | xpcomf1o.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
| 5 | f1oeq1 6836 | . . . 4 ⊢ (𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) → (𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) ↔ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵))) | |
| 6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) ↔ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵)) |
| 7 | 3, 6 | mpbir 231 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) |
| 8 | cnvxp 6177 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
| 9 | f1oeq3 6838 | . . 3 ⊢ (◡(𝐴 × 𝐵) = (𝐵 × 𝐴) → (𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴))) | |
| 10 | 8, 9 | ax-mp 5 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)) |
| 11 | 7, 10 | mpbi 230 | 1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 Rel wrel 5690 –1-1-onto→wf1o 6560 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: xpcomco 9102 xpcomen 9103 omf1o 9115 swapf1f1o 48981 swapf2f1o 48982 swapf2f1oaALT 48984 |
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