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Mirrors > Home > MPE Home > Th. List > isof1oopb | Structured version Visualization version GIF version |
Description: A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.) |
Ref | Expression |
---|---|
isof1oopb | ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V | |
2 | fvex 6658 | . . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V | |
3 | 1, 2 | opelvv 5558 | . . . . . . . 8 ⊢ 〈(𝐻‘𝑥), (𝐻‘𝑦)〉 ∈ (V × V) |
4 | df-br 5031 | . . . . . . . 8 ⊢ ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) ↔ 〈(𝐻‘𝑥), (𝐻‘𝑦)〉 ∈ (V × V)) | |
5 | 3, 4 | mpbir 234 | . . . . . . 7 ⊢ (𝐻‘𝑥)(V × V)(𝐻‘𝑦) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥(V × V)𝑦 → (𝐻‘𝑥)(V × V)(𝐻‘𝑦)) |
7 | opelvvg 5559 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × V)) | |
8 | df-br 5031 | . . . . . . . 8 ⊢ (𝑥(V × V)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V × V)) | |
9 | 7, 8 | sylibr 237 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥(V × V)𝑦) |
10 | 9 | a1d 25 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) → 𝑥(V × V)𝑦)) |
11 | 6, 10 | impbid2 229 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
12 | 11 | adantl 485 | . . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
13 | 12 | ralrimivva 3156 | . . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
14 | 13 | pm4.71i 563 | . 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))) |
15 | df-isom 6333 | . 2 ⊢ (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))) | |
16 | 14, 15 | bitr4i 281 | 1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 〈cop 4531 class class class wbr 5030 × cxp 5517 –1-1-onto→wf1o 6323 ‘cfv 6324 Isom wiso 6325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-iota 6283 df-fv 6332 df-isom 6333 |
This theorem is referenced by: (None) |
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