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Theorem algrflem 6325

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 5957	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 6253	. . . 4 ⊢ 1^st :V–onto→V
3		fof 5507	. . . 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 4277	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐵, 𝐶⟩ ∈ V)
8	5, 6, 7	mp2an 426	. . 3 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
9		fvco3 5660	. . 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 6242	. . 3 ⊢ (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵
12	11	fveq2i 5589	. 2 ⊢ (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵)
13	1, 10, 12	3eqtri 2231	1 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵)

Colors of variables: wff set class

Syntax hints: = wceq 1373 ∈ wcel 2177 Vcvv 2773 ⟨cop 3638 ∘ ccom 4684 ⟶wf 5273 –onto→wfo 5275 ‘cfv 5277 (class class class)co 5954 1^st c1st 6234

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4167 ax-pow 4223 ax-pr 4258 ax-un 4485

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ral 2490 df-rex 2491 df-v 2775 df-sbc 3001 df-un 3172 df-in 3174 df-ss 3181 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-br 4049 df-opab 4111 df-mpt 4112 df-id 4345 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-fo 5283 df-fv 5285 df-ov 5957 df-1st 6236

This theorem is referenced by: algrf 12417

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