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Theorem xpcomf1o 8990

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 5641	. . . 4 ⊢ Rel (𝐴 × 𝐵)
2		cnvf1o 8051	. . . 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 6756	. . . 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 6110	. . 3 ⊢ ^◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
9		f1oeq3 6758	. . 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 1540 {csn 4579 ∪ cuni 4861 ↦ cmpt 5176 × cxp 5621 ^◡ccnv 5622 Rel wrel 5628 –1-1-onto→wf1o 6485

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-1st 7931 df-2nd 7932

This theorem is referenced by: xpcomco 8991 xpcomen 8992 omf1o 9004 swapf1f1o 49264 swapf2f1o 49265 swapf2f1oaALT 49267