fncoOLD - Metamath Proof Explorer

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Theorem fncoOLD 6668

Description: Obsolete version of fnco 6667 as of 20-Sep-2024. (Contributed by NM, 22-May-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
fncoOLD	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

**Proof of Theorem fncoOLD**
Step	Hyp	Ref	Expression
1		fnfun 6649	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fnfun 6649	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
3		funco 6588	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2an 595	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺))
5	4	3adant3 1131	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → Fun (𝐹 ∘ 𝐺))
6		fndm 6652	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	sseq2d 4014	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺 ⊆ 𝐴))
8	7	biimpar 477	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → ran 𝐺 ⊆ dom 𝐹)
9		dmcosseq 5972	. . . . 5 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺)
10	8, 9	syl 17	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
11	10	3adant2 1130	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
12		fndm 6652	. . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13	12	3ad2ant2 1133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵)
14	11, 13	eqtrd 2771	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = 𝐵)
15		df-fn 6546	. 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵))
16	5, 14, 15	sylanbrc 582	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ⊆ wss 3948 dom cdm 5676 ran crn 5677 ∘ ccom 5680 Fun wfun 6537 Fn wfn 6538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-fun 6545 df-fn 6546

This theorem is referenced by: (None)

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