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Mirrors > Home > NFE Home > Th. List > rabxp | GIF version |
Description: Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.) |
Ref | Expression |
---|---|
rabxp.1 | ⊢ (x = 〈y, z〉 → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rabxp | ⊢ {x ∈ (A × B) ∣ φ} = {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B ∧ ψ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4801 | . . . . 5 ⊢ (x ∈ (A × B) ↔ ∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B))) | |
2 | 1 | anbi1i 676 | . . . 4 ⊢ ((x ∈ (A × B) ∧ φ) ↔ (∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B)) ∧ φ)) |
3 | 19.41vv 1902 | . . . 4 ⊢ (∃y∃z((x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ (∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B)) ∧ φ)) | |
4 | anass 630 | . . . . . 6 ⊢ (((x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ (x = 〈y, z〉 ∧ ((y ∈ A ∧ z ∈ B) ∧ φ))) | |
5 | rabxp.1 | . . . . . . . . 9 ⊢ (x = 〈y, z〉 → (φ ↔ ψ)) | |
6 | 5 | anbi2d 684 | . . . . . . . 8 ⊢ (x = 〈y, z〉 → (((y ∈ A ∧ z ∈ B) ∧ φ) ↔ ((y ∈ A ∧ z ∈ B) ∧ ψ))) |
7 | df-3an 936 | . . . . . . . 8 ⊢ ((y ∈ A ∧ z ∈ B ∧ ψ) ↔ ((y ∈ A ∧ z ∈ B) ∧ ψ)) | |
8 | 6, 7 | syl6bbr 254 | . . . . . . 7 ⊢ (x = 〈y, z〉 → (((y ∈ A ∧ z ∈ B) ∧ φ) ↔ (y ∈ A ∧ z ∈ B ∧ ψ))) |
9 | 8 | pm5.32i 618 | . . . . . 6 ⊢ ((x = 〈y, z〉 ∧ ((y ∈ A ∧ z ∈ B) ∧ φ)) ↔ (x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B ∧ ψ))) |
10 | 4, 9 | bitri 240 | . . . . 5 ⊢ (((x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ (x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B ∧ ψ))) |
11 | 10 | 2exbii 1583 | . . . 4 ⊢ (∃y∃z((x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ ∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B ∧ ψ))) |
12 | 2, 3, 11 | 3bitr2i 264 | . . 3 ⊢ ((x ∈ (A × B) ∧ φ) ↔ ∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B ∧ ψ))) |
13 | 12 | abbii 2465 | . 2 ⊢ {x ∣ (x ∈ (A × B) ∧ φ)} = {x ∣ ∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B ∧ ψ))} |
14 | df-rab 2623 | . 2 ⊢ {x ∈ (A × B) ∣ φ} = {x ∣ (x ∈ (A × B) ∧ φ)} | |
15 | df-opab 4623 | . 2 ⊢ {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B ∧ ψ)} = {x ∣ ∃y∃z(x = 〈y, z〉 ∧ (y ∈ A ∧ z ∈ B ∧ ψ))} | |
16 | 13, 14, 15 | 3eqtr4i 2383 | 1 ⊢ {x ∈ (A × B) ∣ φ} = {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B ∧ ψ)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 {crab 2618 〈cop 4561 {copab 4622 × cxp 4770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-xp 4784 |
This theorem is referenced by: (None) |
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