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Theorem foco 6832

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

**Assertion**
Ref	Expression
foco	⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → (𝐹 ∘ 𝐺):𝐴⟶onto𝐶)

**Proof of Theorem foco**
Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐹:𝐵⟶onto𝐶)
2		fofun 6819	. . . 4 ⊢ (𝐺:𝐴⟶onto𝐵 → Fun 𝐺)
3	2	adantl 487	. . 3 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → Fun 𝐺)
4		forn 6821	. . . . 5 ⊢ (𝐺:𝐴⟶onto𝐵 → ran 𝐺 = 𝐵)
5		eqimss2 3972	. . . . 5 ⊢ (ran 𝐺 = 𝐵 → 𝐵 ⊆ ran 𝐺)
6	4, 5	syl 18	. . . 4 ⊢ (𝐺:𝐴⟶onto𝐵 → 𝐵 ⊆ ran 𝐺)
7	6	adantl 487	. . 3 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐵 ⊆ ran 𝐺)
8		focofo 6831	. . 3 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ Fun 𝐺 ∧ 𝐵 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)⟶onto𝐶)
9	1, 3, 7, 8	syl3anc 1399	. 2 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)⟶onto𝐶)
10		focnvimacdmdm 6830	. . . . 5 ⊢ (𝐺:𝐴⟶onto𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
11	10	eqcomd 2769	. . . 4 ⊢ (𝐺:𝐴⟶onto𝐵 → 𝐴 = (^◡𝐺 “ 𝐵))
12	11	adantl 487	. . 3 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐴 = (^◡𝐺 “ 𝐵))
13		foeq2 6815	. . 3 ⊢ (𝐴 = (^◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶onto𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)⟶onto𝐶))
14	12, 13	syl 18	. 2 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶onto𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)⟶onto𝐶))
15	9, 14	mpbird 260	1 ⊢ ((𝐹:𝐵⟶onto𝐶 ∧ 𝐺:𝐴⟶onto𝐵) → (𝐹 ∘ 𝐺):𝐴⟶onto𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ⊆ wss 3900 ^◡ccnv 5654 ran crn 5656 “ cima 5658 ∘ ccom 5659 Fun wfun 6552 ⟶ontowfo 6556

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-pr 5395

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 2102 df-mo 2567 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-br 5107 df-opab 5171 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-fun 6560 df-fn 6561 df-f 6562 df-fo 6564

This theorem is used by: f1oco 6870 wdomtr 9596 fin1a2lem7 10511 cofull 18140 sursubmefmnd 19121 uniiccdif 25929 fcoresfob 48347

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