Theorem foco 5399

Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

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		dffo2 5393	. . 3 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶))
2		dffo2 5393	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵))
3		fco 5332	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
4	3	ad2ant2r 501	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
5		fdm 5322	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → dom 𝐹 = 𝐵)
6		eqtr3 2177	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
7	5, 6	sylan 281	. . . . . . 7 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8		rncoeq 4856	. . . . . . . . 9 ⊢ (dom 𝐹 = ran 𝐺 → ran (𝐹 ∘ 𝐺) = ran 𝐹)
9	8	eqeq1d 2166	. . . . . . . 8 ⊢ (dom 𝐹 = ran 𝐺 → (ran (𝐹 ∘ 𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
10	9	biimpar 295	. . . . . . 7 ⊢ ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
11	7, 10	sylan 281	. . . . . 6 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
12	11	an32s 558	. . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹 ∘ 𝐺) = 𝐶)
13	12	adantrl 470	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹 ∘ 𝐺) = 𝐶)
14	4, 13	jca 304	. . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
15	1, 2, 14	syl2anb 289	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
16		dffo2 5393	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
17	15, 16	sylibr 133	1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335  dom cdm 4583  ran crn 4584   ∘ ccom 4587  ⟶wf 5163  –onto→wfo 5165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-fun 5169  df-fn 5170  df-f 5171  df-fo 5173

This theorem is referenced by:  f1oco  5434  ennnfonelemnn0  12123  ctinfomlemom  12128  qnnen  12132  enctlem  12133