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Theorem xpcomf1o 9055

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 5681	. . . 4 ⊢ Rel (𝐴 × 𝐵)
2		cnvf1o 8107	. . . 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 6810	. . . 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 234	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→^◡(𝐴 × 𝐵)
8		cnvxp 6156	. . 3 ⊢ ^◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
9		f1oeq3 6812	. . 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 233	1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 = wceq 1570 {csn 4590 ∪ cuni 4873 ↦ cmpt 5193 × cxp 5661 ^◡ccnv 5662 Rel wrel 5668 –1-1-onto→wf1o 6537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-1st 7987 df-2nd 7988

This theorem is referenced by: xpcomco 9056 xpcomen 9057 omf1o 9069 swapf1f1o 50036 swapf2f1o 50037 swapf2f1oaALT 50039