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Theorem xpcomf1o 9006

Description: The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.)

**Hypothesis**
Ref	Expression
xpcomf1o.1	⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ^◡{𝑥})

**Assertion**
Ref	Expression
xpcomf1o	⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 Allowed substitution hint: 𝐹(𝑥)

**Proof of Theorem xpcomf1o**
Step	Hyp	Ref	Expression
1		relxp 5650	. . . 4 ⊢ Rel (𝐴 × 𝐵)
2		cnvf1o 8063	. . . 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 6770	. . . 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 6123	. . 3 ⊢ ^◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
9		f1oeq3 6772	. . 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 1542 {csn 4582 ∪ cuni 4865 ↦ cmpt 5181 × cxp 5630 ^◡ccnv 5631 Rel wrel 5637 –1-1-onto→wf1o 6499

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: xpcomco 9007 xpcomen 9008 omf1o 9020 swapf1f1o 49631 swapf2f1o 49632 swapf2f1oaALT 49634