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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 5853 | . 2 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) | |
2 | fo1st 6133 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fof 5418 | . . . 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 4211 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 〈𝐵, 𝐶〉 ∈ V) | |
8 | 5, 6, 7 | mp2an 424 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ V |
9 | fvco3 5565 | . . 3 ⊢ ((1st :V⟶V ∧ 〈𝐵, 𝐶〉 ∈ V) → ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉))) | |
10 | 4, 8, 9 | mp2an 424 | . 2 ⊢ ((𝐹 ∘ 1st )‘〈𝐵, 𝐶〉) = (𝐹‘(1st ‘〈𝐵, 𝐶〉)) |
11 | 5, 6 | op1st 6122 | . . 3 ⊢ (1st ‘〈𝐵, 𝐶〉) = 𝐵 |
12 | 11 | fveq2i 5497 | . 2 ⊢ (𝐹‘(1st ‘〈𝐵, 𝐶〉)) = (𝐹‘𝐵) |
13 | 1, 10, 12 | 3eqtri 2195 | 1 ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 Vcvv 2730 〈cop 3584 ∘ ccom 4613 ⟶wf 5192 –onto→wfo 5194 ‘cfv 5196 (class class class)co 5850 1st c1st 6114 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fo 5202 df-fv 5204 df-ov 5853 df-1st 6116 |
This theorem is referenced by: algrf 11986 |
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