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Mirrors > Home > ILE Home > Th. List > algrflem | GIF version |
Description: Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
algrflem.1 | ⊢ 𝐵 ∈ V |
algrflem.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
algrflem | ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5770 | . 2 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) | |
2 | fo1st 6048 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fof 5340 | . . . 4 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1st :V⟶V |
5 | algrflem.1 | . . . 4 ⊢ 𝐵 ∈ V | |
6 | algrflem.2 | . . . 4 ⊢ 𝐶 ∈ V | |
7 | opexg 4145 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 〈𝐵, 𝐶〉 ∈ V) | |
8 | 5, 6, 7 | mp2an 422 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ V |
9 | fvco3 5485 | . . 3 ⊢ ((1st :V⟶V ∧ 〈𝐵, 𝐶〉 ∈ V) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) | |
10 | 4, 8, 9 | mp2an 422 | . 2 ⊢ ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉)) |
11 | 5, 6 | op1st 6037 | . . 3 ⊢ (1st ‘〈𝐵, 𝐶〉) = 𝐵 |
12 | 11 | fveq2i 5417 | . 2 ⊢ (𝐹‘(1st ‘〈𝐵, 𝐶〉)) = (𝐹‘𝐵) |
13 | 1, 10, 12 | 3eqtri 2162 | 1 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 〈cop 3525 ∘ ccom 4538 ⟶wf 5114 –onto→wfo 5116 ‘cfv 5118 (class class class)co 5767 1st c1st 6029 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-ov 5770 df-1st 6031 |
This theorem is referenced by: algrf 11715 |
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