Theorem foco2 5426

Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Assertion

Ref	Expression
foco2	⊢ ((F:B–→C ∧ G:A–→B ∧ (F ∘ G):A–onto→C) → F:B–onto→C)

Proof of Theorem foco2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 955	. 2 ⊢ ((F:B–→C ∧ G:A–→B ∧ (F ∘ G):A–onto→C) → F:B–→C)
2		foelrn 5425	. . . . . 6 ⊢ (((F ∘ G):A–onto→C ∧ y ∈ C) → ∃z ∈ A y = ((F ∘ G) ‘z))
3		ffvelrn 5415	. . . . . . . . . 10 ⊢ ((G:A–→B ∧ z ∈ A) → (G ‘z) ∈ B)
4	3	adantll 694	. . . . . . . . 9 ⊢ (((F:B–→C ∧ G:A–→B) ∧ z ∈ A) → (G ‘z) ∈ B)
5		fvco3 5384	. . . . . . . . . 10 ⊢ ((G:A–→B ∧ z ∈ A) → ((F ∘ G) ‘z) = (F ‘(G ‘z)))
6	5	adantll 694	. . . . . . . . 9 ⊢ (((F:B–→C ∧ G:A–→B) ∧ z ∈ A) → ((F ∘ G) ‘z) = (F ‘(G ‘z)))
7		fveq2 5328	. . . . . . . . . . 11 ⊢ (x = (G ‘z) → (F ‘x) = (F ‘(G ‘z)))
8	7	eqeq2d 2364	. . . . . . . . . 10 ⊢ (x = (G ‘z) → (((F ∘ G) ‘z) = (F ‘x) ↔ ((F ∘ G) ‘z) = (F ‘(G ‘z))))
9	8	rspcev 2955	. . . . . . . . 9 ⊢ (((G ‘z) ∈ B ∧ ((F ∘ G) ‘z) = (F ‘(G ‘z))) → ∃x ∈ B ((F ∘ G) ‘z) = (F ‘x))
10	4, 6, 9	syl2anc 642	. . . . . . . 8 ⊢ (((F:B–→C ∧ G:A–→B) ∧ z ∈ A) → ∃x ∈ B ((F ∘ G) ‘z) = (F ‘x))
11		eqeq1 2359	. . . . . . . . 9 ⊢ (y = ((F ∘ G) ‘z) → (y = (F ‘x) ↔ ((F ∘ G) ‘z) = (F ‘x)))
12	11	rexbidv 2635	. . . . . . . 8 ⊢ (y = ((F ∘ G) ‘z) → (∃x ∈ B y = (F ‘x) ↔ ∃x ∈ B ((F ∘ G) ‘z) = (F ‘x)))
13	10, 12	syl5ibrcom 213	. . . . . . 7 ⊢ (((F:B–→C ∧ G:A–→B) ∧ z ∈ A) → (y = ((F ∘ G) ‘z) → ∃x ∈ B y = (F ‘x)))
14	13	rexlimdva 2738	. . . . . 6 ⊢ ((F:B–→C ∧ G:A–→B) → (∃z ∈ A y = ((F ∘ G) ‘z) → ∃x ∈ B y = (F ‘x)))
15	2, 14	syl5 28	. . . . 5 ⊢ ((F:B–→C ∧ G:A–→B) → (((F ∘ G):A–onto→C ∧ y ∈ C) → ∃x ∈ B y = (F ‘x)))
16	15	impl 603	. . . 4 ⊢ ((((F:B–→C ∧ G:A–→B) ∧ (F ∘ G):A–onto→C) ∧ y ∈ C) → ∃x ∈ B y = (F ‘x))
17	16	ralrimiva 2697	. . 3 ⊢ (((F:B–→C ∧ G:A–→B) ∧ (F ∘ G):A–onto→C) → ∀y ∈ C ∃x ∈ B y = (F ‘x))
18	17	3impa 1146	. 2 ⊢ ((F:B–→C ∧ G:A–→B ∧ (F ∘ G):A–onto→C) → ∀y ∈ C ∃x ∈ B y = (F ‘x))
19		dffo3 5422	. 2 ⊢ (F:B–onto→C ↔ (F:B–→C ∧ ∀y ∈ C ∃x ∈ B y = (F ‘x)))
20	1, 18, 19	sylanbrc 645	1 ⊢ ((F:B–→C ∧ G:A–→B ∧ (F ∘ G):A–onto→C) → F:B–onto→C)

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ∘ ccom 4721  –→wf 4777  –onto→wfo 4779   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795

This theorem is referenced by: (None)