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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 6905 | . . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V | |
2 | fvex 6905 | . . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V | |
3 | 1, 2 | opelvv 5717 | . . . . . . . 8 ⊢ ⟨(𝐻‘𝑥), (𝐻‘𝑦)⟩ ∈ (V × V) |
4 | df-br 5150 | . . . . . . . 8 ⊢ ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) ↔ ⟨(𝐻‘𝑥), (𝐻‘𝑦)⟩ ∈ (V × V)) | |
5 | 3, 4 | mpbir 230 | . . . . . . 7 ⊢ (𝐻‘𝑥)(V × V)(𝐻‘𝑦) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥(V × V)𝑦 → (𝐻‘𝑥)(V × V)(𝐻‘𝑦)) |
7 | opelvvg 5718 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V)) | |
8 | df-br 5150 | . . . . . . . 8 ⊢ (𝑥(V × V)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)) | |
9 | 7, 8 | sylibr 233 | . . . . . . 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 225 | . . . . 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 3199 | . . 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 559 | . 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 6553 | . 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 277 | 1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ⟨cop 4635 class class class wbr 5149 × cxp 5675 –1-1-onto→wf1o 6543 ‘cfv 6544 Isom wiso 6545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-iota 6496 df-fv 6552 df-isom 6553 |
This theorem is referenced by: (None) |
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