Theorem fcoresfo 47841

Description: If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-2024.)

Hypotheses

Ref	Expression
fcores.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcores.e	⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
fcores.p	⊢ 𝑃 = (◡𝐹 “ 𝐶)
fcores.x	⊢ 𝑋 = (𝐹 ↾ 𝑃)
fcores.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
fcores.y	⊢ 𝑌 = (𝐺 ↾ 𝐸)
fcoresfo.s	⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)

Assertion

Ref	Expression
fcoresfo	⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)

Proof of Theorem fcoresfo

Step	Hyp	Ref	Expression
1		fcores.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
2		fcores.e	. . . . . 6 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐶))
4		inss2 4190	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
5	3, 4	eqsstrdi 3982	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
6	1, 5	fssresd 6749	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷)
7		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
8	7	feq1i 6700	. . 3 ⊢ (𝑌:𝐸⟶𝐷 ↔ (𝐺 ↾ 𝐸):𝐸⟶𝐷)
9	6, 8	sylibr 237	. 2 ⊢ (𝜑 → 𝑌:𝐸⟶𝐷)
10		fcores.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
11		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
12		fcores.x	. . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
13	10, 2, 11, 12	fcoreslem3 47835	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
14		fof 6796	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
15	13, 14	syl 18	. 2 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
16		fcoresfo.s	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
17	10, 2, 11, 12, 1, 7	fcores 47837	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
18	17	eqcomd 2771	. . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹))
19		foeq1 6792	. . . 4 ⊢ ((𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹) → ((𝑌 ∘ 𝑋):𝑃–onto→𝐷 ↔ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
20	18, 19	syl 18	. . 3 ⊢ (𝜑 → ((𝑌 ∘ 𝑋):𝑃–onto→𝐷 ↔ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
21	16, 20	mpbird 260	. 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋):𝑃–onto→𝐷)
22		foco2 7108	. 2 ⊢ ((𝑌:𝐸⟶𝐷 ∧ 𝑋:𝑃⟶𝐸 ∧ (𝑌 ∘ 𝑋):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
23	9, 15, 21, 22	syl3anc 1398	1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∩ cin 3905  ^◡ccnv 5662  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667  ⟶wf 6536  –onto→wfo 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  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 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548

This theorem is used by:  fcoresfob  47842  funfocofob  47848