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Mirrors > Home > MPE Home > Th. List > fococnv2 | Structured version Visualization version GIF version |
Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Ref | Expression |
---|---|
fococnv2 | ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 6566 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funcocnv2 6614 | . . 3 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
4 | forn 6568 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
5 | 4 | reseq2d 5818 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵)) |
6 | 3, 5 | eqtrd 2833 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 I cid 5424 ◡ccnv 5518 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 Fun wfun 6318 –onto→wfo 6322 |
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-rab 3115 df-v 3443 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-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 |
This theorem is referenced by: f1ococnv2 6616 foeqcnvco 7034 |
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