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Theorem xpcomf1o 9057

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 5693	. . . 4 ⊢ Rel (𝐴 × 𝐵)
2		cnvf1o 8093	. . . 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 6818	. . . 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 230	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→^◡(𝐴 × 𝐵)
8		cnvxp 6153	. . 3 ⊢ ^◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
9		f1oeq3 6820	. . 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 229	1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1541 {csn 4627 ∪ cuni 4907 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 Rel wrel 5680 –1-1-onto→wf1o 6539

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1st 7971 df-2nd 7972

This theorem is referenced by: xpcomco 9058 xpcomen 9059 omf1o 9071