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Mirrors > Home > MPE Home > Th. List > fncofn | Structured version Visualization version GIF version |
Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6672. (Contributed by AV, 17-Sep-2024.) |
Ref | Expression |
---|---|
fncofn | ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 6654 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | funco 6593 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
3 | 1, 2 | sylan 579 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) |
4 | 3 | funfnd 6584 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺)) |
5 | fndm 6657 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴) |
7 | 6 | eqcomd 2734 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹) |
8 | 7 | imaeq2d 6063 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (◡𝐺 “ 𝐴) = (◡𝐺 “ dom 𝐹)) |
9 | dmco 6258 | . . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹) | |
10 | 8, 9 | eqtr4di 2786 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (◡𝐺 “ 𝐴) = dom (𝐹 ∘ 𝐺)) |
11 | 10 | fneq2d 6648 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ↔ (𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺))) |
12 | 4, 11 | mpbird 257 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ◡ccnv 5677 dom cdm 5678 “ cima 5681 ∘ ccom 5682 Fun wfun 6542 Fn wfn 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6550 df-fn 6551 |
This theorem is referenced by: fnco 6672 fcof 6746 |
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