Theorem omf1o 9064

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 2769	. . . . . 6 ⊢ (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
2	1	omxpenlem 9062	. . . . 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 463	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
4		eqid 2769	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})
5	4	xpcomf1o 9050	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6		f1oco 6842	. . . 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 597	. . 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 9051	. . . . 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 2795	. . . 4 ⊢ 𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}))
11		f1oeq1 6806	. . . 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 237	. 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 9062	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16		f1ocnv 6831	. . 3 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) → ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴))
17	15, 16	syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴))
18		f1oco 6842	. 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 595	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {csn 4591  ∪ cuni 4873   ↦ cmpt 5193   × cxp 5657  ^◡ccnv 5658   ∘ ccom 5663  Oncon0 6357  –1-1-onto→wf1o 6532  (class class class)co 7408   ∈ cmpo 7410   +_o coa 8446   ·_o comu 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-omul 8454

This theorem is referenced by:  cnfcom3  9669  infxpenc  9998