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Mirrors > Home > MPE Home > Th. List > funcoeqres | Structured version Visualization version GIF version |
Description: Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
funcoeqres | ⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcocnv2 6690 | . . . 4 ⊢ (Fun 𝐺 → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺)) | |
2 | 1 | coeq2d 5736 | . . 3 ⊢ (Fun 𝐺 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺))) |
3 | coass 6134 | . . . 4 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺)) | |
4 | 3 | eqcomi 2746 | . . 3 ⊢ (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = ((𝐹 ∘ 𝐺) ∘ ◡𝐺) |
5 | coires1 6133 | . . 3 ⊢ (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺) | |
6 | 2, 4, 5 | 3eqtr3g 2801 | . 2 ⊢ (Fun 𝐺 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺)) |
7 | coeq1 5731 | . 2 ⊢ ((𝐹 ∘ 𝐺) = 𝐻 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐻 ∘ ◡𝐺)) | |
8 | 6, 7 | sylan9req 2799 | 1 ⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 I cid 5459 ◡ccnv 5555 ran crn 5557 ↾ cres 5558 ∘ ccom 5560 Fun wfun 6379 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 |
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 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-sn 4547 df-pr 4549 df-op 4553 df-br 5059 df-opab 5121 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-fun 6387 |
This theorem is referenced by: frlmup4 20768 evlseu 21048 |
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