Theorem foco 5057

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 5051	. . 3 ⊢ (𝐹:B–onto→𝐶 ↔ (𝐹:B⟶𝐶 ∧ ran 𝐹 = 𝐶))
2		dffo2 5051	. . 3 ⊢ (𝐺:A–onto→B ↔ (𝐺:A⟶B ∧ ran 𝐺 = B))
3		fco 4997	. . . . 5 ⊢ ((𝐹:B⟶𝐶 ∧ 𝐺:A⟶B) → (𝐹 ∘ 𝐺):A⟶𝐶)
4	3	ad2ant2r 478	. . . 4 ⊢ (((𝐹:B⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:A⟶B ∧ ran 𝐺 = B)) → (𝐹 ∘ 𝐺):A⟶𝐶)
5		fdm 4991	. . . . . . . 8 ⊢ (𝐹:B⟶𝐶 → dom 𝐹 = B)
6		eqtr3 2056	. . . . . . . 8 ⊢ ((dom 𝐹 = B ∧ ran 𝐺 = B) → dom 𝐹 = ran 𝐺)
7	5, 6	sylan 267	. . . . . . 7 ⊢ ((𝐹:B⟶𝐶 ∧ ran 𝐺 = B) → dom 𝐹 = ran 𝐺)
8		rncoeq 4547	. . . . . . . . 9 ⊢ (dom 𝐹 = ran 𝐺 → ran (𝐹 ∘ 𝐺) = ran 𝐹)
9	8	eqeq1d 2045	. . . . . . . 8 ⊢ (dom 𝐹 = ran 𝐺 → (ran (𝐹 ∘ 𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
10	9	biimpar 281	. . . . . . 7 ⊢ ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
11	7, 10	sylan 267	. . . . . 6 ⊢ (((𝐹:B⟶𝐶 ∧ ran 𝐺 = B) ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
12	11	an32s 502	. . . . 5 ⊢ (((𝐹:B⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = B) → ran (𝐹 ∘ 𝐺) = 𝐶)
13	12	adantrl 447	. . . 4 ⊢ (((𝐹:B⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:A⟶B ∧ ran 𝐺 = B)) → ran (𝐹 ∘ 𝐺) = 𝐶)
14	4, 13	jca 290	. . 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 275	. 2 ⊢ ((𝐹:B–onto→𝐶 ∧ 𝐺:A–onto→B) → ((𝐹 ∘ 𝐺):A⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
16		dffo2 5051	. 2 ⊢ ((𝐹 ∘ 𝐺):A–onto→𝐶 ↔ ((𝐹 ∘ 𝐺):A⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
17	15, 16	sylibr 137	1 ⊢ ((𝐹:B–onto→𝐶 ∧ 𝐺:A–onto→B) → (𝐹 ∘ 𝐺):A–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  dom cdm 4287  ran crn 4288   ∘ ccom 4291  ⟶wf 4840  –onto→wfo 4842

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-fun 4846  df-fn 4847  df-f 4848  df-fo 4850

This theorem is referenced by:  f1oco  5090