![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5785 | . 2 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) | |
2 | fo1st 6063 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fof 5353 | . . . 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 4158 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 〈𝐵, 𝐶〉 ∈ V) | |
8 | 5, 6, 7 | mp2an 423 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ V |
9 | fvco3 5500 | . . 3 ⊢ ((1st :V⟶V ∧ 〈𝐵, 𝐶〉 ∈ V) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) | |
10 | 4, 8, 9 | mp2an 423 | . 2 ⊢ ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉)) |
11 | 5, 6 | op1st 6052 | . . 3 ⊢ (1st ‘〈𝐵, 𝐶〉) = 𝐵 |
12 | 11 | fveq2i 5432 | . 2 ⊢ (𝐹‘(1st ‘〈𝐵, 𝐶〉)) = (𝐹‘𝐵) |
13 | 1, 10, 12 | 3eqtri 2165 | 1 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 Vcvv 2689 〈cop 3535 ∘ ccom 4551 ⟶wf 5127 –onto→wfo 5129 ‘cfv 5131 (class class class)co 5782 1st c1st 6044 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-ov 5785 df-1st 6046 |
This theorem is referenced by: algrf 11762 |
Copyright terms: Public domain | W3C validator |