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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 5227 | . . . 4 ⊢ Rel (𝐴 × 𝐵) | |
2 | cnvf1o 7276 | . . . 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 6127 | . . . 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 221 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) |
8 | cnvxp 5551 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
9 | f1oeq3 6129 | . . 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 220 | 1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 {csn 4177 ∪ cuni 4436 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 Rel wrel 5119 –1-1-onto→wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: xpcomco 8050 xpcomen 8051 omf1o 8063 |
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