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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 6678 | . . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V | |
2 | fvex 6678 | . . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V | |
3 | 1, 2 | opelvv 5589 | . . . . . . . 8 ⊢ 〈(𝐻‘𝑥), (𝐻‘𝑦)〉 ∈ (V × V) |
4 | df-br 5060 | . . . . . . . 8 ⊢ ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) ↔ 〈(𝐻‘𝑥), (𝐻‘𝑦)〉 ∈ (V × V)) | |
5 | 3, 4 | mpbir 233 | . . . . . . 7 ⊢ (𝐻‘𝑥)(V × V)(𝐻‘𝑦) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥(V × V)𝑦 → (𝐻‘𝑥)(V × V)(𝐻‘𝑦)) |
7 | opelvvg 5590 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × V)) | |
8 | df-br 5060 | . . . . . . . 8 ⊢ (𝑥(V × V)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V × V)) | |
9 | 7, 8 | sylibr 236 | . . . . . . 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 228 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
12 | 11 | adantl 484 | . . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))) |
13 | 12 | ralrimivva 3191 | . . 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 562 | . 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 6359 | . 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 280 | 1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 〈cop 4567 class class class wbr 5059 × cxp 5548 –1-1-onto→wf1o 6349 ‘cfv 6350 Isom wiso 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-iota 6309 df-fv 6358 df-isom 6359 |
This theorem is referenced by: (None) |
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