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Theorem foco 6848

Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.)

**Assertion**
Ref	Expression
foco	⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

**Proof of Theorem foco**
Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐵–onto→𝐶)
2		fofun 6835	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → Fun 𝐺)
3	2	adantl 481	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐺)
4		forn 6837	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
5		eqimss2 4068	. . . . 5 ⊢ (ran 𝐺 = 𝐵 → 𝐵 ⊆ ran 𝐺)
6	4, 5	syl 17	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 ⊆ ran 𝐺)
7	6	adantl 481	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐵 ⊆ ran 𝐺)
8		focofo 6847	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ Fun 𝐺 ∧ 𝐵 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
9	1, 3, 7, 8	syl3anc 1371	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
10		focnvimacdmdm 6846	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
11	10	eqcomd 2746	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐴 = (^◡𝐺 “ 𝐵))
12	11	adantl 481	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 = (^◡𝐺 “ 𝐵))
13		foeq2 6831	. . 3 ⊢ (𝐴 = (^◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶))
14	12, 13	syl 17	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶))
15	9, 14	mpbird 257	1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3976 ^◡ccnv 5699 ran crn 5701 “ cima 5703 ∘ ccom 5704 Fun wfun 6567 –onto→wfo 6571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579

This theorem is referenced by: f1oco 6885 wdomtr 9644 fin1a2lem7 10475 cofull 18001 sursubmefmnd 18931 uniiccdif 25632 fcoresfob 46987

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