fcoreslem4 - Mathbox for Alexander van der Vekens

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Theorem fcoreslem4 47076

Description: Lemma 4 for fcores 47077. (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 6648	. . . 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 3977	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
7	2, 6	fnssresd 6601	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9	8	fneq1i 6574	. . 3 ⊢ (𝑌 Fn 𝐸 ↔ (𝐺 ↾ 𝐸) Fn 𝐸)
10	7, 9	sylibr 234	. 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 47075	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
15		fofn 6733	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → 𝑋 Fn 𝑃)
17	11, 3, 12, 13	fcoreslem2 47074	. . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸)
18		eqimss 3991	. . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸)
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸)
20		fnco 6595	. 2 ⊢ ((𝑌 Fn 𝐸 ∧ 𝑋 Fn 𝑃 ∧ ran 𝑋 ⊆ 𝐸) → (𝑌 ∘ 𝑋) Fn 𝑃)
21	10, 16, 19, 20	syl3anc 1373	1 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∩ cin 3899 ⊆ wss 3900 ^◡ccnv 5613 ran crn 5615 ↾ cres 5616 “ cima 5617 ∘ ccom 5618 Fn wfn 6472 ⟶wf 6473 –onto→wfo 6475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rab 3394 df-v 3436 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-br 5090 df-opab 5152 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-fun 6479 df-fn 6480 df-f 6481 df-fo 6483

This theorem is referenced by: fcores 47077

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