Theorem foco2 7129

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 7127	. . . . . 6 ⊢ (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧))
2		ffvelcdm 7101	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐵)
3		fvco3 7008	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
4		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑧)))
5	4	rspceeqv 3645	. . . . . . . . 9 ⊢ (((𝐺‘𝑧) ∈ 𝐵 ∧ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
6	2, 3, 5	syl2anc 584	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
7		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (𝑦 = (𝐹‘𝑥) ↔ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
8	7	rexbidv 3177	. . . . . . . 8 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
9	6, 8	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
10	9	rexlimdva 3153	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
11	1, 10	syl5 34	. . . . 5 ⊢ (𝐺:𝐴⟶𝐵 → (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
12	11	impl 455	. . . 4 ⊢ (((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
13	12	ralrimiva 3144	. . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
14	13	anim2i 617	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
15		3anass 1094	. 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 7122	. 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
17	14, 15, 16	3imtr4i 292	1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∘ ccom 5693  ⟶wf 6559  –onto→wfo 6561  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571

This theorem is referenced by:  fcoresfo  47021