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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 6565 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | 1 | simprbi 496 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
3 | iss 6055 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
4 | dfdm4 5909 | . . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹 | |
5 | dmcoeq 5992 | . . . . . . 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 5700 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
8 | 6, 7 | eqtr4i 2766 | . . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
9 | 8 | reseq2i 5997 | . . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹) |
10 | 9 | eqeq2i 2748 | . . 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 1537 ⊆ wss 3963 I cid 5582 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 Rel wrel 5694 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-fun 6565 |
This theorem is referenced by: fococnv2 6875 f1cocnv2 6877 funcoeqres 6880 fcoinver 32624 tocyc01 33121 cocnv 37712 frege131d 43754 gricushgr 47824 |
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