fcoreslem4 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem4		Structured version Visualization version GIF version

Theorem fcoreslem4 44538

Description: Lemma 4 for fcores 44539. (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 6593	. . . 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 4163	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶
6	4, 5	eqsstrdi 3974	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶)
7	2, 6	fnssresd 6548	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9	8	fneq1i 6522	. . 3 ⊢ (𝑌 Fn 𝐸 ↔ (𝐺 ↾ 𝐸) Fn 𝐸)
10	7, 9	sylibr 233	. 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 44537	. . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
15		fofn 6682	. . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → 𝑋 Fn 𝑃)
17	11, 3, 12, 13	fcoreslem2 44536	. . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸)
18		eqimss 3976	. . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸)
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸)
20		fnco 6541	. 2 ⊢ ((𝑌 Fn 𝐸 ∧ 𝑋 Fn 𝑃 ∧ ran 𝑋 ⊆ 𝐸) → (𝑌 ∘ 𝑋) Fn 𝑃)
21	10, 16, 19, 20	syl3anc 1370	1 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∩ cin 3885 ⊆ wss 3886 ^◡ccnv 5583 ran crn 5585 ↾ cres 5586 “ cima 5587 ∘ ccom 5588 Fn wfn 6421 ⟶wf 6422 –onto→wfo 6424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5074 df-opab 5136 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-fun 6428 df-fn 6429 df-f 6430 df-fo 6432

This theorem is referenced by: fcores 44539

W3C validator