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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 5573 | . . . 4 ⊢ Rel (𝐴 × 𝐵) | |
2 | cnvf1o 7806 | . . . 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 6604 | . . . 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 233 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵) |
8 | cnvxp 6014 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
9 | f1oeq3 6606 | . . 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 232 | 1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 {csn 4567 ∪ cuni 4838 ↦ cmpt 5146 × cxp 5553 ◡ccnv 5554 Rel wrel 5560 –1-1-onto→wf1o 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: xpcomco 8607 xpcomen 8608 omf1o 8620 |
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