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Theorem algrflemg 6079

Description: Lemma for algrf 11566 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 5729	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 6007	. . . . 5 ⊢ 1^st :V–onto→V
3		fof 5301	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 7	. . . 4 ⊢ 1^st :V⟶V
5		opexg 4108	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6		fvco3 5444	. . . 4 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	sylancr 408	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
8		op1stg 6000	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵)
9	8	fveq2d 5377	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵))
10	7, 9	eqtrd 2145	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘𝐵))
11	1, 10	syl5eq 2157	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∈ wcel 1461 Vcvv 2655 ⟨cop 3494 ∘ ccom 4501 ⟶wf 5075 –onto→wfo 5077 ‘cfv 5079 (class class class)co 5726 1^st c1st 5988

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313

This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fo 5085 df-fv 5087 df-ov 5729 df-1st 5990

This theorem is referenced by: ialgrlem1st 11563 algrp1 11567

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