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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 2610 | . . . . . 6 ⊢ (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) = (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) | |
2 | 1 | omxpenlem 7946 | . . . . 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 468 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴)) |
4 | eqid 2610 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) | |
5 | 4 | xpcomf1o 7934 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵) |
6 | f1oco 6072 | . . . 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 693 | . . 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 7935 | . . . . 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 2635 | . . . 4 ⊢ 𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})) |
11 | f1oeq1 6040 | . . . 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 223 | . 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 7946 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵)) |
16 | f1ocnv 6062 | . . 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 6072 | . 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 691 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 ∪ cuni 4372 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 ∘ ccom 5042 Oncon0 5640 –1-1-onto→wf1o 5803 (class class class)co 6549 ↦ cmpt2 6551 +𝑜 coa 7444 ·𝑜 comu 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 |
This theorem is referenced by: cnfcom3 8484 infxpenc 8724 |
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