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Theorem algrflemg 6209

Description: Lemma for algrf 11999 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 5856	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 6136	. . . . 5 ⊢ 1^st :V–onto→V
3		fof 5420	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
5		opexg 4213	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6		fvco3 5567	. . . 4 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	sylancr 412	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
8		op1stg 6129	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵)
9	8	fveq2d 5500	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵))
10	7, 9	eqtrd 2203	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘𝐵))
11	1, 10	eqtrid 2215	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 1348 ∈ wcel 2141 Vcvv 2730 ⟨cop 3586 ∘ ccom 4615 ⟶wf 5194 –onto→wfo 5196 ‘cfv 5198 (class class class)co 5853 1^st c1st 6117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 df-ov 5856 df-1st 6119

This theorem is referenced by: ialgrlem1st 11996 algrp1 12000

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