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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 6563 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
| 2 | 1 | simprbi 496 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
| 3 | iss 6053 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
| 4 | dfdm4 5906 | . . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹 | |
| 5 | dmcoeq 5989 | . . . . . . 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 5696 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
| 8 | 6, 7 | eqtr4i 2768 | . . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
| 9 | 8 | reseq2i 5994 | . . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹) |
| 10 | 9 | eqeq2i 2750 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
| 11 | 3, 10 | bitri 275 | . 2 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
| 12 | 2, 11 | sylib 218 | 1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 Fun wfun 6555 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-fun 6563 |
| This theorem is referenced by: fococnv2 6874 f1cocnv2 6876 funcoeqres 6879 fcoinver 32617 tocyc01 33138 cocnv 37732 frege131d 43777 gricushgr 47886 |
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