Theorem omf1o 9015

Description: Construct an explicit bijection from 𝐴 ·_o 𝐵 to 𝐵 ·_o 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
omf1o.1	⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
omf1o.2	⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))

Assertion

Ref	Expression
omf1o	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem omf1o

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . . . 6 ⊢ (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
2	1	omxpenlem 9013	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
3	2	ancoms 459	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
4		eqid 2740	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})
5	4	xpcomf1o 9001	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6		f1oco 6797	. . . 4 ⊢ (((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴) ∧ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)) → ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
7	3, 5, 6	sylancl 592	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
8		omf1o.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
9	4, 1	xpcomco 9002	. . . . 5 ⊢ ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
10	8, 9	eqtr4i 2766	. . . 4 ⊢ 𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}))
11		f1oeq1 6762	. . . 4 ⊢ (𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})) → (𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴) ↔ ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴)))
12	10, 11	ax-mp 5	. . 3 ⊢ (𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴) ↔ ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
13	7, 12	sylibr 235	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
14		omf1o.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
15	14	omxpenlem 9013	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16		f1ocnv 6786	. . 3 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) → ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴))
17	15, 16	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴))
18		f1oco 6797	. 2 ⊢ ((𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴) ∧ ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴)) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
19	13, 17, 18	syl2anc 590	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4562  ∪ cuni 4845   ↦ cmpt 5160   × cxp 5623  ^◡ccnv 5624   ∘ ccom 5629  Oncon0 6317  –1-1-onto→wf1o 6491  (class class class)co 7363   ∈ cmpo 7365   +_o coa 8399   ·_o comu 8400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-omul 8407

This theorem is referenced by:  cnfcom3  9623  infxpenc  9938