Theorem foco2 7106

Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)

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		foelrn 7104	. . . . . 6 ⊢ (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧))
2		ffvelcdm 7078	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐵)
3		fvco3 6983	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
4		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑧)))
5	4	rspceeqv 3605	. . . . . . . . 9 ⊢ (((𝐺‘𝑧) ∈ 𝐵 ∧ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
6	2, 3, 5	syl2anc 595	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
7		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (𝑦 = (𝐹‘𝑥) ↔ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
8	7	rexbidv 3189	. . . . . . . 8 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
9	6, 8	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
10	9	rexlimdva 3166	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
11	1, 10	syl5 35	. . . . 5 ⊢ (𝐺:𝐴⟶𝐵 → (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
12	11	impl 460	. . . 4 ⊢ (((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
13	12	ralrimiva 3157	. . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
14	13	anim2i 628	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
15		3anass 1111	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)))
16		dffo3 7099	. 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
17	14, 15, 16	3imtr4i 295	1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∘ ccom 5667  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546

This theorem is referenced by:  fcoresfo  47785