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Theorem algrflemg 5851

Description: Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)

**Assertion**
Ref	Expression
algrflemg	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

**Proof of Theorem algrflemg**
Step	Hyp	Ref	Expression
1		df-ov 5515	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 5784	. . . . 5 ⊢ 1^st :V–onto→V
3		fof 5106	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 7	. . . 4 ⊢ 1^st :V⟶V
5		opexg 3964	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6		fvco3 5244	. . . 4 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	sylancr 393	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
8		op1stg 5777	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵)
9	8	fveq2d 5182	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵))
10	7, 9	eqtrd 2072	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘𝐵))
11	1, 10	syl5eq 2084	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ⟨cop 3378 ∘ ccom 4349 ⟶wf 4898 –onto→wfo 4900 ‘cfv 4902 (class class class)co 5512 1^st c1st 5765

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-ov 5515 df-1st 5767

This theorem is referenced by: ialgrlem1st 9881 ialgrp1 9885

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