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Theorem algrflem 7870

Description: Lemma for algrf 16093 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 7194	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 7759	. . . 4 ⊢ 1^st :V–onto→V
3		fof 6611	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1^st :V⟶V
5		opex 5333	. . 3 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
6		fvco3 6788	. . 3 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	mp2an 692	. 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 7747	. . 3 ⊢ (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵
11	10	fveq2i 6698	. 2 ⊢ (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵)
12	1, 7, 11	3eqtri 2763	1 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵)

Colors of variables: wff setvar class

Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 ⟨cop 4533 ∘ ccom 5540 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 1^st c1st 7737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-ov 7194 df-1st 7739

This theorem is referenced by: fpwwe 10225 seq1st 16091 algrf 16093 algrp1 16094 dvnff 24774 dvnp1 24776

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