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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 6701 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfsbc1v 3455 | . . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 | |
3 | 2 | 19.41 2103 | . . . 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 7220 | . . . . . . . 8 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 ↔ 𝜑)) | |
5 | 4 | pm5.32i 669 | . . . . . . 7 ⊢ ((𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
6 | 5 | exbii 1774 | . . . . . 6 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
7 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
8 | nfsbc1v 3455 | . . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑤) / 𝑦]𝜑 | |
9 | 7, 8 | nfsbc 3457 | . . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 |
10 | 9 | 19.41 2103 | . . . . . 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 266 | . . . . 5 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
12 | 11 | exbii 1774 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = 〈𝑥, 𝑦〉 ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
13 | elvv 5177 | . . . . 5 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉) | |
14 | 13 | anbi1i 731 | . . . 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 292 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)) |
16 | 15 | opabbii 4717 | . 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 2644 | 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 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 〈cop 4183 {copab 4712 × cxp 5112 ‘cfv 5888 {coprab 6651 1st c1st 7166 2nd c2nd 7167 |
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-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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-iota 5851 df-fun 5890 df-fv 5896 df-oprab 6654 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: dfoprab3 7224 |
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