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Mirrors > Home > ILE Home > Th. List > algrflemg | GIF version |
Description: Lemma for algrf 11956 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 5839 | . 2 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) | |
2 | fo1st 6117 | . . . . 5 ⊢ 1st :V–onto→V | |
3 | fof 5404 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
5 | opexg 4200 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐵, 𝐶〉 ∈ V) | |
6 | fvco3 5551 | . . . 4 ⊢ ((1st :V⟶V ∧ 〈𝐵, 𝐶〉 ∈ V) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) | |
7 | 4, 5, 6 | sylancr 411 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) |
8 | op1stg 6110 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (1st ‘〈𝐵, 𝐶〉) = 𝐵) | |
9 | 8 | fveq2d 5484 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘(1st ‘〈𝐵, 𝐶〉)) = (𝐹‘𝐵)) |
10 | 7, 9 | eqtrd 2197 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘𝐵)) |
11 | 1, 10 | syl5eq 2209 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 〈cop 3573 ∘ ccom 4602 ⟶wf 5178 –onto→wfo 5180 ‘cfv 5182 (class class class)co 5836 1st c1st 6098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-ov 5839 df-1st 6100 |
This theorem is referenced by: ialgrlem1st 11953 algrp1 11957 |
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