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Mirrors > Home > ILE Home > Th. List > algrflemg | GIF version |
Description: Lemma for algrf 12025 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.) |
Ref | Expression |
---|---|
algrflemg | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5872 | . 2 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) | |
2 | fo1st 6152 | . . . . 5 ⊢ 1st :V–onto→V | |
3 | fof 5434 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
5 | opexg 4225 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐵, 𝐶〉 ∈ V) | |
6 | fvco3 5583 | . . . 4 ⊢ ((1st :V⟶V ∧ 〈𝐵, 𝐶〉 ∈ V) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) | |
7 | 4, 5, 6 | sylancr 414 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) |
8 | op1stg 6145 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1st ‘〈𝐵, 𝐶〉) = 𝐵) | |
9 | 8 | fveq2d 5515 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1st ‘〈𝐵, 𝐶〉)) = (𝐹‘𝐵)) |
10 | 7, 9 | eqtrd 2210 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘𝐵)) |
11 | 1, 10 | eqtrid 2222 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3594 ∘ ccom 4627 ⟶wf 5208 –onto→wfo 5210 ‘cfv 5212 (class class class)co 5869 1st c1st 6133 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fo 5218 df-fv 5220 df-ov 5872 df-1st 6135 |
This theorem is referenced by: ialgrlem1st 12022 algrp1 12026 |
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