fcoreslem4 - Mathbox for Alexander van der Vekens

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Theorem fcoreslem4 47962

Description: Lemma 4 for fcores 47963. (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)

**Assertion**
Ref	Expression
fcoreslem4	⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)

**Proof of Theorem fcoreslem4**
Step	Hyp	Ref	Expression
1		fcores.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
2	1	ffnd 6707	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐶)
3		fcores.e	. . . . . 6 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐶))
5		inss2 4186	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
6	4, 5	eqsstrdi 3978	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
7	2, 6	fnssresd 6660	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9	8	fneq1i 6633	. . 3 ⊢ (𝑌 Fn 𝐸 ↔ (𝐺 ↾ 𝐸) Fn 𝐸)
10	7, 9	sylibr 237	. 2 ⊢ (𝜑 → 𝑌 Fn 𝐸)
11		fcores.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12		fcores.p	. . . 4 ⊢ 𝑃 = (^◡𝐹 “ 𝐶)
13		fcores.x	. . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
14	11, 3, 12, 13	fcoreslem3 47961	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
15		fofn 6795	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃)
16	14, 15	syl 18	. 2 ⊢ (𝜑 → 𝑋 Fn 𝑃)
17	11, 3, 12, 13	fcoreslem2 47960	. . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸)
18		eqimss 3992	. . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸)
19	17, 18	syl 18	. 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸)
20		fnco 6654	. 2 ⊢ ((𝑌 Fn 𝐸 ∧ 𝑋 Fn 𝑃 ∧ ran 𝑋 ⊆ 𝐸) → (𝑌 ∘ 𝑋) Fn 𝑃)
21	10, 16, 19, 20	syl3anc 1398	1 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∩ cin 3901 ⊆ wss 3902 ^◡ccnv 5658 ran crn 5660 ↾ cres 5661 “ cima 5662 ∘ ccom 5663 Fn wfn 6532 ⟶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 2215 ax-ext 2734 ax-sep 5255 ax-pr 5402

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-br 5108 df-opab 5172 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-fun 6539 df-fn 6540 df-f 6541 df-fo 6543

This theorem is used by: fcores 47963

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