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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 6357 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | 1 | simprbi 499 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
3 | iss 5903 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
4 | dfdm4 5764 | . . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹 | |
5 | dmcoeq 5845 | . . . . . . 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 5566 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
8 | 6, 7 | eqtr4i 2847 | . . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
9 | 8 | reseq2i 5850 | . . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹) |
10 | 9 | eqeq2i 2834 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
11 | 3, 10 | bitri 277 | . 2 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
12 | 2, 11 | sylib 220 | 1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3936 I cid 5459 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 Rel wrel 5560 Fun wfun 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-fun 6357 |
This theorem is referenced by: fococnv2 6640 f1cocnv2 6642 funcoeqres 6645 fcoinver 30357 tocyc01 30760 cocnv 35015 frege131d 40129 isomushgr 44011 |
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