Theorem fcoresfo 48110

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 4183	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
5	3, 4	eqsstrdi 3975	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
6	1, 5	fssresd 6747	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷)
7		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
8	7	feq1i 6698	. . 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 48104	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
14		fof 6794	. . 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 48106	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
18	17	eqcomd 2767	. . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹))
19		foeq1 6790	. . . 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 7107	. 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 3898  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  ⟶wf 6533  –onto→wfo 6535

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545

This theorem is used by:  fcoresfob  48111  funfocofob  48117