fcoresf1lem - Mathbox for Alexander van der Vekens

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Theorem fcoresf1lem 45764

Description: Lemma for fcoresf1 45765. (Contributed by AV, 18-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
fcoresf1lem	⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))

**Proof of Theorem fcoresf1lem**
Step	Hyp	Ref	Expression
1		fcores.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		fcores.e	. . . . 5 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
3		fcores.p	. . . . 5 ⊢ 𝑃 = (^◡𝐹 “ 𝐶)
4		fcores.x	. . . . 5 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
5		fcores.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
6		fcores.y	. . . . 5 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
7	1, 2, 3, 4, 5, 6	fcores 45763	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
8	7	fveq1d 6890	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍))
9	8	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍))
10	1, 2, 3, 4	fcoreslem3 45761	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
11		fof 6802	. . . . 5 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
13	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
14		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑍 ∈ 𝑃)
15	13, 14	fvco3d 6988	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
16	9, 15	eqtrd 2772	1 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ^◡ccnv 5674 ran crn 5676 ↾ cres 5677 “ cima 5678 ∘ ccom 5679 ⟶wf 6536 –onto→wfo 6538 ‘cfv 6540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548

This theorem is referenced by: fcoresf1 45765

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