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Theorem algrflemg 6426

Description: Lemma for algrf 12742 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised 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 6053	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 6351	. . . . 5 ⊢ 1^st :V–onto→V
3		fof 5590	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
5		opexg 4344	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6		fvco3 5748	. . . 4 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	sylancr 414	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
8		op1stg 6344	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵)
9	8	fveq2d 5674	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵))
10	7, 9	eqtrd 2265	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘𝐵))
11	1, 10	eqtrid 2277	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2203 Vcvv 2813 ⟨cop 3692 ∘ ccom 4753 ⟶wf 5348 –onto→wfo 5350 ‘cfv 5352 (class class class)co 6050 1^st c1st 6332

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2815 df-sbc 3043 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fo 5358 df-fv 5360 df-ov 6053 df-1st 6334

This theorem is referenced by: ialgrlem1st 12739 algrp1 12743

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