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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 6113 | . . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V | |
2 | fvex 6113 | . . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V | |
3 | 1, 2 | opelvv 5088 | . . . . . . . 8 ⊢ 〈(𝐻‘𝑥), (𝐻‘𝑦)〉 ∈ (V × V) |
4 | df-br 4584 | . . . . . . . 8 ⊢ ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) ↔ 〈(𝐻‘𝑥), (𝐻‘𝑦)〉 ∈ (V × V)) | |
5 | 3, 4 | mpbir 220 | . . . . . . 7 ⊢ (𝐻‘𝑥)(V × V)(𝐻‘𝑦) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥(V × V)𝑦 → (𝐻‘𝑥)(V × V)(𝐻‘𝑦)) |
7 | opelvvg 5087 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × V)) | |
8 | df-br 4584 | . . . . . . . 8 ⊢ (𝑥(V × V)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V × V)) | |
9 | 7, 8 | sylibr 223 | . . . . . . 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 215 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
13 | 12 | ralrimivva 2954 | . . 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 662 | . 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 5813 | . 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 266 | 1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 〈cop 4131 class class class wbr 4583 × cxp 5036 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-iota 5768 df-fv 5812 df-isom 5813 |
This theorem is referenced by: (None) |
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