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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnresfnco | Structured version Visualization version GIF version |
Description: Composition of two functions, similar to fnco 6687. (Contributed by Alexander van der Vekens, 25-Jul-2017.) |
Ref | Expression |
---|---|
fnresfnco | ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 6669 | . . 3 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺)) | |
2 | fnfun 6669 | . . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
3 | funresfunco 6609 | . . 3 ⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺)) |
5 | fndm 6672 | . . . . . 6 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺) | |
6 | dmres 6032 | . . . . . . . 8 ⊢ dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹) | |
7 | 6 | eqeq1i 2740 | . . . . . . 7 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺) |
8 | dfss2 3981 | . . . . . . 7 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺) | |
9 | 7, 8 | sylbb2 238 | . . . . . 6 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 → ran 𝐺 ⊆ dom 𝐹) |
10 | 5, 9 | syl 17 | . . . . 5 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → ran 𝐺 ⊆ dom 𝐹) |
11 | 10 | adantr 480 | . . . 4 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → ran 𝐺 ⊆ dom 𝐹) |
12 | dmcosseq 5990 | . . . 4 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
14 | fndm 6672 | . . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
15 | 14 | adantl 481 | . . 3 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom 𝐺 = 𝐵) |
16 | 13, 15 | eqtrd 2775 | . 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = 𝐵) |
17 | df-fn 6566 | . 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵)) | |
18 | 4, 16, 17 | sylanbrc 583 | 1 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 dom cdm 5689 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 Fun wfun 6557 Fn wfn 6558 |
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-11 2155 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-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 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 df-fn 6566 |
This theorem is referenced by: funcoressn 46992 |
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