Theorem fcoresfo 45771

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 4229	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
5	3, 4	eqsstrdi 4036	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
6	1, 5	fssresd 6758	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷)
7		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
8	7	feq1i 6708	. . 3 ⊢ (𝑌:𝐸⟶𝐷 ↔ (𝐺 ↾ 𝐸):𝐸⟶𝐷)
9	6, 8	sylibr 233	. 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 45765	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
14		fof 6805	. . 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 45767	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
18	17	eqcomd 2738	. . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹))
19		foeq1 6801	. . . 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 256	. 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 1371	1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∩ cin 3947  ^◡ccnv 5675  ran crn 5677   ↾ cres 5678   “ cima 5679   ∘ ccom 5680  ⟶wf 6539  –onto→wfo 6541

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551

This theorem is referenced by:  fcoresfob  45772  funfocofob  45776