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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 6584 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funcocnv2 6632 | . . 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 6586 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
5 | 4 | reseq2d 5846 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵)) |
6 | 3, 5 | eqtrd 2855 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 I cid 5452 ◡ccnv 5547 ran crn 5549 ↾ cres 5550 ∘ ccom 5552 Fun wfun 6342 –onto→wfo 6346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 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-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 |
This theorem is referenced by: f1ococnv2 6634 foeqcnvco 7049 |
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