fcoresf1lem - Mathbox for Alexander van der Vekens

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Theorem fcoresf1lem 46983

Description: Lemma for fcoresf1 46984. (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 46982	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
8	7	fveq1d 6922	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍))
9	8	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍))
10	1, 2, 3, 4	fcoreslem3 46980	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
11		fof 6834	. . . . 5 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
14		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑍 ∈ 𝑃)
15	13, 14	fvco3d 7022	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
16	9, 15	eqtrd 2780	1 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ^◡ccnv 5699 ran crn 5701 ↾ cres 5702 “ cima 5703 ∘ ccom 5704 ⟶wf 6569 –onto→wfo 6571 ‘cfv 6573

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579 df-fv 6581

This theorem is referenced by: fcoresf1 46984

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