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Theorem fcoOLD 6621

Description: Obsolete version of fco 6620 as of 20-Sep-2024. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

**Assertion**
Ref	Expression
fcoOLD	⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)

**Proof of Theorem fcoOLD**
Step	Hyp	Ref	Expression
1		df-f 6434	. . 3 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
2		df-f 6434	. . 3 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))
3		fnco 6545	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)
4	3	3expib 1120	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴))
5	4	adantr 480	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴))
6		rncoss 5878	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
7		sstr 3933	. . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶)
8	6, 7	mpan 686	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐶 → ran (𝐹 ∘ 𝐺) ⊆ 𝐶)
9	8	adantl 481	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶)
10	5, 9	jctird 526	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)))
11	10	imp 406	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))
12	1, 2, 11	syl2anb 597	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))
13		df-f 6434	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ↔ ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))
14	12, 13	sylibr 233	1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3891 ran crn 5589 ∘ ccom 5592 Fn wfn 6425 ⟶wf 6426

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-br 5079 df-opab 5141 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-fun 6432 df-fn 6433 df-f 6434

This theorem is referenced by: (None)

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