Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6652 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funcocnv2 6703 | . . 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 6654 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
5 | 4 | reseq2d 5865 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵)) |
6 | 3, 5 | eqtrd 2778 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 I cid 5468 ◡ccnv 5564 ran crn 5566 ↾ cres 5567 ∘ ccom 5569 Fun wfun 6391 –onto→wfo 6395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-br 5068 df-opab 5130 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-fun 6399 df-fn 6400 df-f 6401 df-fo 6403 |
This theorem is referenced by: f1ococnv2 6705 foeqcnvco 7128 |
Copyright terms: Public domain | W3C validator |