Theorem foco2 7063

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 7061	. . . . . 6 ⊢ (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧))
2		ffvelcdm 7035	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐵)
3		fvco3 6941	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
4		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑧)))
5	4	rspceeqv 3601	. . . . . . . . 9 ⊢ (((𝐺‘𝑧) ∈ 𝐵 ∧ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
6	2, 3, 5	syl2anc 585	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
7		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (𝑦 = (𝐹‘𝑥) ↔ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
8	7	rexbidv 3162	. . . . . . . 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 3139	. . . . . 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 3130	. . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
14	13	anim2i 618	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
15		3anass 1095	. 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 7056	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∘ ccom 5636  ⟶wf 6496  –onto→wfo 6498  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508

This theorem is referenced by:  fcoresfo  47425