fcoresf1lem - Mathbox for Alexander van der Vekens

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Theorem fcoresf1lem 47772

Description: Lemma for fcoresf1 47773. (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 47771	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
8	7	fveq1d 6883	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍))
9	8	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍))
10	1, 2, 3, 4	fcoreslem3 47769	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
11		fof 6792	. . . . 5 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
13	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
14		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑍 ∈ 𝑃)
15	13, 14	fvco3d 6982	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
16	9, 15	eqtrd 2796	1 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1568 ∈ wcel 2141 ∩ cin 3903 ^◡ccnv 5660 ran crn 5662 ↾ cres 5663 “ cima 5664 ∘ ccom 5665 ⟶wf 6532 –onto→wfo 6534 ‘cfv 6536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544

This theorem is referenced by: fcoresf1 47773

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