fcoreslem4 - Mathbox for Alexander van der Vekens

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Theorem fcoreslem4 48080

Description: Lemma 4 for fcores 48081. (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 6702	. . . 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 4183	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
6	4, 5	eqsstrdi 3975	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
7	2, 6	fnssresd 6655	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9	8	fneq1i 6628	. . 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 48079	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
15		fofn 6790	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃)
16	14, 15	syl 18	. 2 ⊢ (𝜑 → 𝑋 Fn 𝑃)
17	11, 3, 12, 13	fcoreslem2 48078	. . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸)
18		eqimss 3989	. . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸)
19	17, 18	syl 18	. 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸)
20		fnco 6649	. 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 3898 ⊆ wss 3899 ^◡ccnv 5650 ran crn 5652 ↾ cres 5653 “ cima 5654 ∘ ccom 5655 Fn wfn 6526 ⟶wf 6527 –onto→wfo 6529

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-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-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 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-br 5104 df-opab 5168 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-fun 6533 df-fn 6534 df-f 6535 df-fo 6537

This theorem is used by: fcores 48081

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