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Mirrors > Home > MPE Home > Th. List > evl1fval1lem | Structured version Visualization version GIF version |
Description: Lemma for evl1fval1 21713. (Contributed by AV, 11-Sep-2019.) |
Ref | Expression |
---|---|
evl1fval1.q | ⊢ 𝑄 = (eval1‘𝑅) |
evl1fval1.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
evl1fval1lem | ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2737 | . . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | evl1fval1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 21710 | . 2 ⊢ (eval1‘𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) |
5 | evl1fval1.q | . . 3 ⊢ 𝑄 = (eval1‘𝑅) | |
6 | 5 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (eval1‘𝑅)) |
7 | 3 | fvexi 6861 | . . . . 5 ⊢ 𝐵 ∈ V |
8 | 7 | pwid 4587 | . . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵 |
9 | eqid 2737 | . . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵) | |
10 | eqid 2737 | . . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅) | |
11 | 9, 10, 3 | evls1fval 21701 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵))) |
12 | 8, 11 | mpan2 690 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵))) |
13 | 2, 3 | evlval 21521 | . . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
14 | 13 | eqcomi 2746 | . . . 4 ⊢ ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅) |
15 | 14 | coeq2i 5821 | . . 3 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) |
16 | 12, 15 | eqtrdi 2793 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))) |
17 | 4, 6, 16 | 3eqtr4a 2803 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 𝒫 cpw 4565 {csn 4591 ↦ cmpt 5193 × cxp 5636 ∘ ccom 5642 ‘cfv 6501 (class class class)co 7362 1oc1o 8410 ↑m cmap 8772 Basecbs 17090 evalSub ces 21496 eval cevl 21497 evalSub1 ces1 21695 eval1ce1 21696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-evls 21498 df-evl 21499 df-evls1 21697 df-evl1 21698 |
This theorem is referenced by: evl1fval1 21713 |
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