Theorem fcoresfo 47088

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 4237	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
5	3, 4	eqsstrdi 4027	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
6	1, 5	fssresd 6774	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷)
7		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
8	7	feq1i 6726	. . 3 ⊢ (𝑌:𝐸⟶𝐷 ↔ (𝐺 ↾ 𝐸):𝐸⟶𝐷)
9	6, 8	sylibr 234	. 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 47082	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
14		fof 6819	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
15	13, 14	syl 17	. 2 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
16		fcoresfo.s	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
17	10, 2, 11, 12, 1, 7	fcores 47084	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
18	17	eqcomd 2742	. . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹))
19		foeq1 6815	. . . 4 ⊢ ((𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹) → ((𝑌 ∘ 𝑋):𝑃–onto→𝐷 ↔ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → ((𝑌 ∘ 𝑋):𝑃–onto→𝐷 ↔ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
21	16, 20	mpbird 257	. 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋):𝑃–onto→𝐷)
22		foco2 7128	. 2 ⊢ ((𝑌:𝐸⟶𝐷 ∧ 𝑋:𝑃⟶𝐸 ∧ (𝑌 ∘ 𝑋):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
23	9, 15, 21, 22	syl3anc 1372	1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∩ cin 3949  ^◡ccnv 5683  ran crn 5685   ↾ cres 5686   “ cima 5687   ∘ ccom 5688  ⟶wf 6556  –onto→wfo 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568

This theorem is referenced by:  fcoresfob  47089  funfocofob  47095