fcoreslem4 - Mathbox for Alexander van der Vekens

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Theorem fcoreslem4 47180

Description: Lemma 4 for fcores 47181. (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 6660	. . . 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 4189	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
6	4, 5	eqsstrdi 3976	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
7	2, 6	fnssresd 6613	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9	8	fneq1i 6586	. . 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 47179	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
15		fofn 6745	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → 𝑋 Fn 𝑃)
17	11, 3, 12, 13	fcoreslem2 47178	. . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸)
18		eqimss 3990	. . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸)
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸)
20		fnco 6607	. 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 3898 ⊆ wss 3899 ^◡ccnv 5620 ran crn 5622 ↾ cres 5623 “ cima 5624 ∘ ccom 5625 Fn wfn 6484 ⟶wf 6485 –onto→wfo 6487

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374

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 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-br 5096 df-opab 5158 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-fun 6491 df-fn 6492 df-f 6493 df-fo 6495

This theorem is referenced by: fcores 47181

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