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Theorem foco 6754

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 6741	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → Fun 𝐺)
3	2	adantl 481	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐺)
4		forn 6743	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
5		eqimss2 3990	. . . . 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 6753	. . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ Fun 𝐺 ∧ 𝐵 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
9	1, 3, 7, 8	syl3anc 1373	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–onto→𝐶)
10		focnvimacdmdm 6752	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
11	10	eqcomd 2739	. . . 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 6737	. . 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 1541 ⊆ wss 3898 ^◡ccnv 5618 ran crn 5620 “ cima 5622 ∘ ccom 5623 Fun wfun 6480 –onto→wfo 6484

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-br 5094 df-opab 5156 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-fun 6488 df-fn 6489 df-f 6490 df-fo 6492

This theorem is referenced by: f1oco 6791 wdomtr 9468 fin1a2lem7 10304 cofull 17845 sursubmefmnd 18806 uniiccdif 25507 fcoresfob 47197

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