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Mirrors > Home > MPE Home > Th. List > funcocnv2 | Structured version Visualization version GIF version |
Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
funcocnv2 | ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 6545 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | 1 | simprbi 495 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
3 | iss 6034 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
4 | dfdm4 5892 | . . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹 | |
5 | dmcoeq 5971 | . . . . . . 7 ⊢ (dom 𝐹 = ran ◡𝐹 → dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹 |
7 | df-rn 5683 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
8 | 6, 7 | eqtr4i 2756 | . . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
9 | 8 | reseq2i 5976 | . . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹) |
10 | 9 | eqeq2i 2738 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
11 | 3, 10 | bitri 274 | . 2 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
12 | 2, 11 | sylib 217 | 1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⊆ wss 3939 I cid 5569 ◡ccnv 5671 dom cdm 5672 ran crn 5673 ↾ cres 5674 ∘ ccom 5676 Rel wrel 5677 Fun wfun 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-fun 6545 |
This theorem is referenced by: fococnv2 6860 f1cocnv2 6862 funcoeqres 6865 fcoinver 32439 tocyc01 32884 cocnv 37255 frege131d 43259 gricushgr 47295 |
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