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Theorem algrflem 6394

Description: Lemma for algrf 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 6021	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 6320	. . . 4 ⊢ 1^st :V–onto→V
3		fof 5559	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1^st :V⟶V
5		algrflem.1	. . . 4 ⊢ 𝐵 ∈ V
6		algrflem.2	. . . 4 ⊢ 𝐶 ∈ V
7		opexg 4320	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐵, 𝐶⟩ ∈ V)
8	5, 6, 7	mp2an 426	. . 3 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
9		fvco3 5717	. . 3 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
10	4, 8, 9	mp2an 426	. 2 ⊢ ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩))
11	5, 6	op1st 6309	. . 3 ⊢ (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵
12	11	fveq2i 5642	. 2 ⊢ (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵)
13	1, 10, 12	3eqtri 2256	1 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵)

Colors of variables: wff set class

Syntax hints: = wceq 1397 ∈ wcel 2202 Vcvv 2802 ⟨cop 3672 ∘ ccom 4729 ⟶wf 5322 –onto→wfo 5324 ‘cfv 5326 (class class class)co 6018 1^st c1st 6301

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-sbc 3032 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fo 5332 df-fv 5334 df-ov 6021 df-1st 6303

This theorem is referenced by: algrf 12622

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