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Mirrors > Home > MPE Home > Th. List > evl1fval1lem | Structured version Visualization version GIF version |
Description: Lemma for evl1fval1 19695. (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 2622 | . . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2622 | . . 3 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅) | |
3 | evl1fval1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 19692 | . 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 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
8 | 3, 7 | eqeltri 2697 | . . . . 5 ⊢ 𝐵 ∈ V |
9 | 8 | pwid 4174 | . . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵 |
10 | eqid 2622 | . . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵) | |
11 | eqid 2622 | . . . . 5 ⊢ (1𝑜 evalSub 𝑅) = (1𝑜 evalSub 𝑅) | |
12 | 10, 11, 3 | evls1fval 19684 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵))) |
13 | 9, 12 | mpan2 707 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵))) |
14 | 2, 3 | evlval 19524 | . . . . 5 ⊢ (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵) |
15 | 14 | eqcomi 2631 | . . . 4 ⊢ ((1𝑜 evalSub 𝑅)‘𝐵) = (1𝑜 eval 𝑅) |
16 | 15 | coeq2i 5282 | . . 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)) |
17 | 13, 16 | syl6eq 2672 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))) |
18 | 4, 6, 17 | 3eqtr4a 2682 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 𝒫 cpw 4158 {csn 4177 ↦ cmpt 4729 × cxp 5112 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 Basecbs 15857 evalSub ces 19504 eval cevl 19505 evalSub1 ces1 19678 eval1ce1 19679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-evls 19506 df-evl 19507 df-evls1 19680 df-evl1 19681 |
This theorem is referenced by: evl1fval1 19695 |
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