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Mirrors > Home > ILE Home > Th. List > dfoprab3s | 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 5494 | . 2 ⊢ {〈〈x, y〉, z〉 ∣ φ} = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} | |
2 | nfsbc1v 2776 | . . . . 5 ⊢ Ⅎx[(1st ‘w) / x][(2nd ‘w) / y]φ | |
3 | 2 | 19.41 1573 | . . . 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 5755 | . . . . . . . 8 ⊢ (w = 〈x, y〉 → ([(1st ‘w) / x][(2nd ‘w) / y]φ ↔ φ)) | |
5 | 4 | pm5.32i 427 | . . . . . . 7 ⊢ ((w = 〈x, y〉 ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ (w = 〈x, y〉 ∧ φ)) |
6 | 5 | exbii 1493 | . . . . . 6 ⊢ (∃y(w = 〈x, y〉 ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ ∃y(w = 〈x, y〉 ∧ φ)) |
7 | nfcv 2175 | . . . . . . . 8 ⊢ Ⅎy(1st ‘w) | |
8 | nfsbc1v 2776 | . . . . . . . 8 ⊢ Ⅎy[(2nd ‘w) / y]φ | |
9 | 7, 8 | nfsbc 2778 | . . . . . . 7 ⊢ Ⅎy[(1st ‘w) / x][(2nd ‘w) / y]φ |
10 | 9 | 19.41 1573 | . . . . . 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 175 | . . . . 5 ⊢ (∃y(w = 〈x, y〉 ∧ φ) ↔ (∃y w = 〈x, y〉 ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)) |
12 | 11 | exbii 1493 | . . . 4 ⊢ (∃x∃y(w = 〈x, y〉 ∧ φ) ↔ ∃x(∃y w = 〈x, y〉 ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)) |
13 | elvv 4345 | . . . . 5 ⊢ (w ∈ (V × V) ↔ ∃x∃y w = 〈x, y〉) | |
14 | 13 | anbi1i 431 | . . . 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 201 | . . 3 ⊢ (∃x∃y(w = 〈x, y〉 ∧ φ) ↔ (w ∈ (V × V) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)) |
16 | 15 | opabbii 3815 | . 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 2057 | 1 ⊢ {〈〈x, y〉, z〉 ∣ φ} = {〈w, z〉 ∣ (w ∈ (V × V) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 [wsbc 2758 〈cop 3370 {copab 3808 × cxp 4286 ‘cfv 4845 {coprab 5456 1st c1st 5707 2nd c2nd 5708 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fv 4853 df-oprab 5459 df-1st 5709 df-2nd 5710 |
This theorem is referenced by: dfoprab3 5759 |
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