Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xpcomf1o | 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 4648 | . . . 4 ⊢ Rel (𝐴 × 𝐵) | |
2 | cnvf1o 6122 | . . . 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 5356 | . . . 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 145 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) |
8 | cnvxp 4957 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
9 | f1oeq3 5358 | . . 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 144 | 1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 {csn 3527 ∪ cuni 3736 ↦ cmpt 3989 × cxp 4537 ◡ccnv 4538 Rel wrel 4544 –1-1-onto→wf1o 5122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: xpcomco 6720 xpcomen 6721 |
Copyright terms: Public domain | W3C validator |