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Mirrors > Home > MPE Home > Th. List > evl1fval1lem | Structured version Visualization version GIF version |
Description: Lemma for evl1fval1 21842. (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 2733 | . . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2733 | . . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | evl1fval1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 21839 | . 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 6903 | . . . . 5 ⊢ 𝐵 ∈ V |
8 | 7 | pwid 4624 | . . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵 |
9 | eqid 2733 | . . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵) | |
10 | eqid 2733 | . . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅) | |
11 | 9, 10, 3 | evls1fval 21830 | . . . 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 21650 | . . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
14 | 13 | eqcomi 2742 | . . . 4 ⊢ ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅) |
15 | 14 | coeq2i 5859 | . . 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 2789 | . 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 2799 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 𝒫 cpw 4602 {csn 4628 ↦ cmpt 5231 × cxp 5674 ∘ ccom 5680 ‘cfv 6541 (class class class)co 7406 1oc1o 8456 ↑m cmap 8817 Basecbs 17141 evalSub ces 21625 eval cevl 21626 evalSub1 ces1 21824 eval1ce1 21825 |
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 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-evls 21627 df-evl 21628 df-evls1 21826 df-evl1 21827 |
This theorem is referenced by: evl1fval1 21842 |
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