algrflemg - Intuitionistic Logic Explorer

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

Theorem algrflemg 5871

Description: Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)

**Assertion**
Ref	Expression
algrflemg	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

**Proof of Theorem algrflemg**
Step	Hyp	Ref	Expression
1		df-ov 5535	. 2 ⊢ (𝐵(𝐹 ∘ 1^st )𝐶) = ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩)
2		fo1st 5804	. . . . 5 ⊢ 1^st :V–onto→V
3		fof 5126	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
4	2, 3	ax-mp 7	. . . 4 ⊢ 1^st :V⟶V
5		opexg 3983	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6		fvco3 5265	. . . 4 ⊢ ((1^st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
7	4, 5, 6	sylancr 405	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)))
8		op1stg 5797	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1^st ‘⟨𝐵, 𝐶⟩) = 𝐵)
9	8	fveq2d 5202	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1^st ‘⟨𝐵, 𝐶⟩)) = (𝐹‘𝐵))
10	7, 9	eqtrd 2113	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1^st )‘⟨𝐵, 𝐶⟩) = (𝐹‘𝐵))
11	1, 10	syl5eq 2125	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⟨cop 3401 ∘ ccom 4367 ⟶wf 4918 –onto→wfo 4920 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-ov 5535 df-1st 5787

This theorem is referenced by: ialgrlem1st 10424 ialgrp1 10428

W3C validator