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Mirrors > Home > NFE Home > Th. List > resoprab | GIF version |
Description: Restriction of an operation class abstraction. (Contributed by set.mm contributors, 10-Feb-2007.) |
Ref | Expression |
---|---|
resoprab | ⊢ ({〈〈x, y〉, z〉 ∣ φ} ↾ (A × B)) = {〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resopab 4999 | . . 3 ⊢ ({〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} ↾ (A × B)) = {〈w, z〉 ∣ (w ∈ (A × B) ∧ ∃x∃y(w = 〈x, y〉 ∧ φ))} | |
2 | 19.42vv 1907 | . . . . 5 ⊢ (∃x∃y(w ∈ (A × B) ∧ (w = 〈x, y〉 ∧ φ)) ↔ (w ∈ (A × B) ∧ ∃x∃y(w = 〈x, y〉 ∧ φ))) | |
3 | an12 772 | . . . . . . 7 ⊢ ((w ∈ (A × B) ∧ (w = 〈x, y〉 ∧ φ)) ↔ (w = 〈x, y〉 ∧ (w ∈ (A × B) ∧ φ))) | |
4 | eleq1 2413 | . . . . . . . . . 10 ⊢ (w = 〈x, y〉 → (w ∈ (A × B) ↔ 〈x, y〉 ∈ (A × B))) | |
5 | opelxp 4811 | . . . . . . . . . 10 ⊢ (〈x, y〉 ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B)) | |
6 | 4, 5 | syl6bb 252 | . . . . . . . . 9 ⊢ (w = 〈x, y〉 → (w ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))) |
7 | 6 | anbi1d 685 | . . . . . . . 8 ⊢ (w = 〈x, y〉 → ((w ∈ (A × B) ∧ φ) ↔ ((x ∈ A ∧ y ∈ B) ∧ φ))) |
8 | 7 | pm5.32i 618 | . . . . . . 7 ⊢ ((w = 〈x, y〉 ∧ (w ∈ (A × B) ∧ φ)) ↔ (w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))) |
9 | 3, 8 | bitri 240 | . . . . . 6 ⊢ ((w ∈ (A × B) ∧ (w = 〈x, y〉 ∧ φ)) ↔ (w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))) |
10 | 9 | 2exbii 1583 | . . . . 5 ⊢ (∃x∃y(w ∈ (A × B) ∧ (w = 〈x, y〉 ∧ φ)) ↔ ∃x∃y(w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))) |
11 | 2, 10 | bitr3i 242 | . . . 4 ⊢ ((w ∈ (A × B) ∧ ∃x∃y(w = 〈x, y〉 ∧ φ)) ↔ ∃x∃y(w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))) |
12 | 11 | opabbii 4626 | . . 3 ⊢ {〈w, z〉 ∣ (w ∈ (A × B) ∧ ∃x∃y(w = 〈x, y〉 ∧ φ))} = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))} |
13 | 1, 12 | eqtri 2373 | . 2 ⊢ ({〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} ↾ (A × B)) = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))} |
14 | dfoprab2 5558 | . . 3 ⊢ {〈〈x, y〉, z〉 ∣ φ} = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} | |
15 | 14 | reseq1i 4930 | . 2 ⊢ ({〈〈x, y〉, z〉 ∣ φ} ↾ (A × B)) = ({〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} ↾ (A × B)) |
16 | dfoprab2 5558 | . 2 ⊢ {〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))} | |
17 | 13, 15, 16 | 3eqtr4i 2383 | 1 ⊢ ({〈〈x, y〉, z〉 ∣ φ} ↾ (A × B)) = {〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 〈cop 4561 {copab 4622 × cxp 4770 ↾ cres 4774 {coprab 5527 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-xp 4784 df-res 4788 df-oprab 5528 |
This theorem is referenced by: resoprab2 5582 |
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