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Mirrors > Home > MPE Home > Th. List > 2fvidf1od | Structured version Visualization version GIF version |
Description: A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.) |
Ref | Expression |
---|---|
2fvcoidd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
2fvcoidd.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
2fvcoidd.i | ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) |
2fvidf1od.i | ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏) |
Ref | Expression |
---|---|
2fvidf1od | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fvcoidd.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 2fvcoidd.g | . 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
3 | 2fvcoidd.i | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) | |
4 | 1, 2, 3 | 2fvcoidd 7044 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) |
5 | 2fvidf1od.i | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏) | |
6 | 2, 1, 5 | 2fvcoidd 7044 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) |
7 | 1, 2, 4, 6 | fcof1od 7041 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∀wral 3135 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: m2cpminv 21296 foresf1o 30192 |
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