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Theorem algrflem 7813

Description: Lemma for algrf 15911 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

**Hypotheses**
Ref	Expression
algrflem.1	⊢ 𝐵 ∈ V
algrflem.2	⊢ 𝐶 ∈ V

**Assertion**
Ref	Expression
algrflem	⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵)

**Proof of Theorem algrflem**
Step	Hyp	Ref	Expression
1		df-ov 7153	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 7703	. . . 4 ⊢ 1^st :V–onto→V
3		fof 6584	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1^st :V⟶V
5		opex 5348	. . 3 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
6		fvco3 6754	. . 3 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	mp2an 690	. 2 ⊢ ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩))
8		algrflem.1	. . . 4 ⊢ 𝐵 ∈ V
9		algrflem.2	. . . 4 ⊢ 𝐶 ∈ V
10	8, 9	op1st 7691	. . 3 ⊢ (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵
11	10	fveq2i 6667	. 2 ⊢ (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵)
12	1, 7, 11	3eqtri 2848	1 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵)

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⟨cop 4566 ∘ ccom 5553 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 1^st c1st 7681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-ov 7153 df-1st 7683

This theorem is referenced by: fpwwe 10062 seq1st 15909 algrf 15911 algrp1 15912 dvnff 24514 dvnp1 24516

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