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Mirrors > Home > MPE Home > Th. List > dfoprab3s | Structured version Visualization version GIF version |
Description: A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfoprab3s | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfoprab2 7206 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfsbc1v 3792 | . . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 | |
3 | 2 | 19.41 2232 | . . . 4 ⊢ (∃𝑥(∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
4 | sbcopeq1a 7742 | . . . . . . . 8 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 ↔ 𝜑)) | |
5 | 4 | pm5.32i 577 | . . . . . . 7 ⊢ ((𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
6 | 5 | exbii 1844 | . . . . . 6 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
7 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
8 | nfsbc1v 3792 | . . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑤) / 𝑦]𝜑 | |
9 | 7, 8 | nfsbcw 3794 | . . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 |
10 | 9 | 19.41 2232 | . . . . . 6 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
11 | 6, 10 | bitr3i 279 | . . . . 5 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
12 | 11 | exbii 1844 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
13 | elvv 5621 | . . . . 5 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉) | |
14 | 13 | anbi1i 625 | . . . 4 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
15 | 3, 12, 14 | 3bitr4i 305 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
16 | 15 | opabbii 5126 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)} |
17 | 1, 16 | eqtri 2844 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 [wsbc 3772 〈cop 4567 {copab 5121 × cxp 5548 ‘cfv 6350 {coprab 7151 1st c1st 7681 2nd c2nd 7682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fv 6358 df-oprab 7154 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: dfoprab3 7746 |
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