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Theorem foco 6807

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 488	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐵–onto→𝐶)
2		fofun 6794	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → Fun 𝐺)
3	2	adantl 487	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐺)
4		forn 6796	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
5		eqimss2 3993	. . . . 5 ⊢ (ran 𝐺 = 𝐵 → 𝐵 ⊆ ran 𝐺)
6	4, 5	syl 18	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 ⊆ ran 𝐺)
7	6	adantl 487	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐵 ⊆ ran 𝐺)
8		focofo 6806	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ Fun 𝐺 ∧ 𝐵 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
9	1, 3, 7, 8	syl3anc 1398	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
10		focnvimacdmdm 6805	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
11	10	eqcomd 2768	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐴 = (^◡𝐺 “ 𝐵))
12	11	adantl 487	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 = (^◡𝐺 “ 𝐵))
13		foeq2 6790	. . 3 ⊢ (𝐴 = (^◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶))
14	12, 13	syl 18	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶))
15	9, 14	mpbird 260	1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ⊆ wss 3902 ^◡ccnv 5658 ran crn 5660 “ cima 5662 ∘ ccom 5663 Fun wfun 6531 –onto→wfo 6535

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-pr 5402

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543

This theorem is used by: f1oco 6845 wdomtr 9551 fin1a2lem7 10412 cofull 18031 sursubmefmnd 19011 uniiccdif 25812 fcoresfob 47968

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