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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 6657. (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 6639 | . . 3 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺)) | |
2 | fnfun 6639 | . . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
3 | funresfunco 6579 | . . 3 ⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2an 595 | . 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺)) |
5 | fndm 6642 | . . . . . 6 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺) | |
6 | dmres 5993 | . . . . . . . 8 ⊢ dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹) | |
7 | 6 | eqeq1i 2729 | . . . . . . 7 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺) |
8 | df-ss 3957 | . . . . . . 7 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺) | |
9 | 7, 8 | sylbb2 237 | . . . . . 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 5962 | . . . 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 6642 | . . . 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 2764 | . 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = 𝐵) |
17 | df-fn 6536 | . 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵)) | |
18 | 4, 16, 17 | sylanbrc 582 | 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 1533 ∩ cin 3939 ⊆ wss 3940 dom cdm 5666 ran crn 5667 ↾ cres 5668 ∘ ccom 5670 Fun wfun 6527 Fn wfn 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-fun 6535 df-fn 6536 |
This theorem is referenced by: funcoressn 46203 |
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