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Mirrors > Home > MPE Home > Th. List > omf1o | Structured version Visualization version GIF version |
Description: Construct an explicit bijection from 𝐴 ·𝑜 𝐵 to 𝐵 ·𝑜 𝐴. (Contributed by Mario Carneiro, 30-May-2015.) |
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)) |
Ref | Expression |
---|---|
omf1o | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . 6 ⊢ (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) = (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) | |
2 | 1 | omxpenlem 8061 | . . . . 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 469 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴)) |
4 | eqid 2622 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) | |
5 | 4 | xpcomf1o 8049 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵) |
6 | f1oco 6159 | . . . 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 694 | . . 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 8050 | . . . . 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 2647 | . . . 4 ⊢ 𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})) |
11 | f1oeq1 6127 | . . . 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 224 | . 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 8061 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵)) |
16 | f1ocnv 6149 | . . 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 6159 | . 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 693 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 ∪ cuni 4436 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 ∘ ccom 5118 Oncon0 5723 –1-1-onto→wf1o 5887 (class class class)co 6650 ↦ cmpt2 6652 +𝑜 coa 7557 ·𝑜 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: cnfcom3 8601 infxpenc 8841 |
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