Theorem fcoresfo 47692

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 4198	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
5	3, 4	eqsstrdi 3989	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
6	1, 5	fssresd 6743	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷)
7		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
8	7	feq1i 6694	. . 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 47686	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
14		fof 6790	. . 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 47688	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
18	17	eqcomd 2775	. . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹))
19		foeq1 6786	. . . 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 7102	. 2 ⊢ ((𝑌:𝐸⟶𝐷 ∧ 𝑋:𝑃⟶𝐸 ∧ (𝑌 ∘ 𝑋):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
23	9, 15, 21, 22	syl3anc 1396	1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∩ cin 3912  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661   “ cima 5662   ∘ ccom 5663  ⟶wf 6530  –onto→wfo 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542

This theorem is referenced by:  fcoresfob  47693  funfocofob  47699