Theorem xpcomf1o 6884

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 4772	. . . 4 ⊢ Rel (𝐴 × 𝐵)
2		cnvf1o 6283	. . . 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 5492	. . . 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 146	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→◡(𝐴 × 𝐵)
8		cnvxp 5088	. . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
9		f1oeq3 5494	. . 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 145	1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1364  {csn 3622  ∪ cuni 3839   ↦ cmpt 4094   × cxp 4661  ^◡ccnv 4662  Rel wrel 4668  –1-1-onto→wf1o 5257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199

This theorem is referenced by:  xpcomco  6885  xpcomen  6886