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Theorem foco 6825

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 6812	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → Fun 𝐺)
3	2	adantl 481	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐺)
4		forn 6814	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
5		eqimss2 4039	. . . . 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 6824	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ Fun 𝐺 ∧ 𝐵 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
9	1, 3, 7, 8	syl3anc 1369	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
10		focnvimacdmdm 6823	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
11	10	eqcomd 2734	. . . 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 6808	. . 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 205 ∧ wa 395 = wceq 1534 ⊆ wss 3947 ^◡ccnv 5677 ran crn 5679 “ cima 5681 ∘ ccom 5682 Fun wfun 6542 –onto→wfo 6546

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 df-f 6552 df-fo 6554

This theorem is referenced by: f1oco 6862 wdomtr 9598 fin1a2lem7 10429 cofull 17922 sursubmefmnd 18847 uniiccdif 25506 fcoresfob 46454

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