Theorem foco 5450

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 5444	. . 3 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶))
2		dffo2 5444	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵))
3		fco 5383	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
4	3	ad2ant2r 509	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
5		fdm 5373	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → dom 𝐹 = 𝐵)
6		eqtr3 2197	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
7	5, 6	sylan 283	. . . . . . 7 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8		rncoeq 4902	. . . . . . . . 9 ⊢ (dom 𝐹 = ran 𝐺 → ran (𝐹 ∘ 𝐺) = ran 𝐹)
9	8	eqeq1d 2186	. . . . . . . 8 ⊢ (dom 𝐹 = ran 𝐺 → (ran (𝐹 ∘ 𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
10	9	biimpar 297	. . . . . . 7 ⊢ ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
11	7, 10	sylan 283	. . . . . 6 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
12	11	an32s 568	. . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹 ∘ 𝐺) = 𝐶)
13	12	adantrl 478	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹 ∘ 𝐺) = 𝐶)
14	4, 13	jca 306	. . 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 291	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
16		dffo2 5444	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
17	15, 16	sylibr 134	1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  dom cdm 4628  ran crn 4629   ∘ ccom 4632  ⟶wf 5214  –onto→wfo 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224

This theorem is referenced by:  f1oco  5486  ennnfonelemnn0  12425  ctinfomlemom  12430  qnnen  12434  enctlem  12435