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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 5909 | . 2 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) | |
2 | fo1st 6197 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fof 5464 | . . . 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 4253 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 〈𝐵, 𝐶〉 ∈ V) | |
8 | 5, 6, 7 | mp2an 426 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ V |
9 | fvco3 5616 | . . 3 ⊢ ((1st :V⟶V ∧ 〈𝐵, 𝐶〉 ∈ V) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) | |
10 | 4, 8, 9 | mp2an 426 | . 2 ⊢ ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉)) |
11 | 5, 6 | op1st 6186 | . . 3 ⊢ (1st ‘〈𝐵, 𝐶〉) = 𝐵 |
12 | 11 | fveq2i 5545 | . 2 ⊢ (𝐹‘(1st ‘〈𝐵, 𝐶〉)) = (𝐹‘𝐵) |
13 | 1, 10, 12 | 3eqtri 2214 | 1 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 Vcvv 2756 〈cop 3617 ∘ ccom 4655 ⟶wf 5238 –onto→wfo 5240 ‘cfv 5242 (class class class)co 5906 1st c1st 6178 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2758 df-sbc 2982 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-fo 5248 df-fv 5250 df-ov 5909 df-1st 6180 |
This theorem is referenced by: algrf 12157 |
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