algrflem - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > algrflem		GIF version

Theorem algrflem 6438

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 6061	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 6364	. . . 4 ⊢ 1^st :V–onto→V
3		fof 5595	. . . 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 4349	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐵, 𝐶⟩ ∈ V)
8	5, 6, 7	mp2an 426	. . 3 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
9		fvco3 5753	. . 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 6353	. . 3 ⊢ (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵
12	11	fveq2i 5678	. 2 ⊢ (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵)
13	1, 10, 12	3eqtri 2259	1 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵)

Colors of variables: wff set class

Syntax hints: = wceq 1398 ∈ wcel 2205 Vcvv 2815 ⟨cop 3697 ∘ ccom 4758 ⟶wf 5353 –onto→wfo 5355 ‘cfv 5357 (class class class)co 6058 1^st c1st 6345

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-v 2817 df-sbc 3046 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fo 5363 df-fv 5365 df-ov 6061 df-1st 6347

This theorem is referenced by: algrf 12767

W3C validator