Theorem foco2 5339

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 938	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵⟶𝐶)
2		foelrn 5338	. . . . . 6 ⊢ (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧))
3		ffvelrn 5321	. . . . . . . . . 10 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐵)
4	3	adantll 459	. . . . . . . . 9 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐵)
5		fvco3 5265	. . . . . . . . . 10 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
6	5	adantll 459	. . . . . . . . 9 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ 𝑧 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
7		fveq2 5198	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑧)))
8	7	eqeq2d 2092	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑧) → (((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥) ↔ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))))
9	8	rspcev 2701	. . . . . . . . 9 ⊢ (((𝐺‘𝑧) ∈ 𝐵 ∧ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
10	4, 6, 9	syl2anc 403	. . . . . . . 8 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
11		eqeq1 2087	. . . . . . . . 9 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (𝑦 = (𝐹‘𝑥) ↔ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
12	11	rexbidv 2369	. . . . . . . 8 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
13	10, 12	syl5ibrcom 155	. . . . . . 7 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
14	13	rexlimdva 2477	. . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
15	2, 14	syl5 32	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
16	15	impl 372	. . . 4 ⊢ ((((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
17	16	ralrimiva 2434	. . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
18	17	3impa 1133	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
19		dffo3 5335	. 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
20	1, 18, 19	sylanbrc 408	1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349   ∘ ccom 4367  ⟶wf 4918  –onto→wfo 4920  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930

This theorem is referenced by: (None)