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Mirrors > Home > NFE Home > Th. List > resoprab2 | GIF version |
Description: Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
resoprab2 | ⊢ ((C ⊆ A ∧ D ⊆ B) → ({〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} ↾ (C × D)) = {〈〈x, y〉, z〉 ∣ ((x ∈ C ∧ y ∈ D) ∧ φ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resoprab 5581 | . 2 ⊢ ({〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} ↾ (C × D)) = {〈〈x, y〉, z〉 ∣ ((x ∈ C ∧ y ∈ D) ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))} | |
2 | anass 630 | . . . 4 ⊢ ((((x ∈ C ∧ y ∈ D) ∧ (x ∈ A ∧ y ∈ B)) ∧ φ) ↔ ((x ∈ C ∧ y ∈ D) ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))) | |
3 | an4 797 | . . . . . 6 ⊢ (((x ∈ C ∧ y ∈ D) ∧ (x ∈ A ∧ y ∈ B)) ↔ ((x ∈ C ∧ x ∈ A) ∧ (y ∈ D ∧ y ∈ B))) | |
4 | ssel 3267 | . . . . . . . . 9 ⊢ (C ⊆ A → (x ∈ C → x ∈ A)) | |
5 | pm4.71 611 | . . . . . . . . 9 ⊢ ((x ∈ C → x ∈ A) ↔ (x ∈ C ↔ (x ∈ C ∧ x ∈ A))) | |
6 | 4, 5 | sylib 188 | . . . . . . . 8 ⊢ (C ⊆ A → (x ∈ C ↔ (x ∈ C ∧ x ∈ A))) |
7 | 6 | bicomd 192 | . . . . . . 7 ⊢ (C ⊆ A → ((x ∈ C ∧ x ∈ A) ↔ x ∈ C)) |
8 | ssel 3267 | . . . . . . . . 9 ⊢ (D ⊆ B → (y ∈ D → y ∈ B)) | |
9 | pm4.71 611 | . . . . . . . . 9 ⊢ ((y ∈ D → y ∈ B) ↔ (y ∈ D ↔ (y ∈ D ∧ y ∈ B))) | |
10 | 8, 9 | sylib 188 | . . . . . . . 8 ⊢ (D ⊆ B → (y ∈ D ↔ (y ∈ D ∧ y ∈ B))) |
11 | 10 | bicomd 192 | . . . . . . 7 ⊢ (D ⊆ B → ((y ∈ D ∧ y ∈ B) ↔ y ∈ D)) |
12 | 7, 11 | bi2anan9 843 | . . . . . 6 ⊢ ((C ⊆ A ∧ D ⊆ B) → (((x ∈ C ∧ x ∈ A) ∧ (y ∈ D ∧ y ∈ B)) ↔ (x ∈ C ∧ y ∈ D))) |
13 | 3, 12 | syl5bb 248 | . . . . 5 ⊢ ((C ⊆ A ∧ D ⊆ B) → (((x ∈ C ∧ y ∈ D) ∧ (x ∈ A ∧ y ∈ B)) ↔ (x ∈ C ∧ y ∈ D))) |
14 | 13 | anbi1d 685 | . . . 4 ⊢ ((C ⊆ A ∧ D ⊆ B) → ((((x ∈ C ∧ y ∈ D) ∧ (x ∈ A ∧ y ∈ B)) ∧ φ) ↔ ((x ∈ C ∧ y ∈ D) ∧ φ))) |
15 | 2, 14 | syl5bbr 250 | . . 3 ⊢ ((C ⊆ A ∧ D ⊆ B) → (((x ∈ C ∧ y ∈ D) ∧ ((x ∈ A ∧ y ∈ B) ∧ φ)) ↔ ((x ∈ C ∧ y ∈ D) ∧ φ))) |
16 | 15 | oprabbidv 5564 | . 2 ⊢ ((C ⊆ A ∧ D ⊆ B) → {〈〈x, y〉, z〉 ∣ ((x ∈ C ∧ y ∈ D) ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))} = {〈〈x, y〉, z〉 ∣ ((x ∈ C ∧ y ∈ D) ∧ φ)}) |
17 | 1, 16 | syl5eq 2397 | 1 ⊢ ((C ⊆ A ∧ D ⊆ B) → ({〈〈x, y〉, z〉 ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} ↾ (C × D)) = {〈〈x, y〉, z〉 ∣ ((x ∈ C ∧ y ∈ D) ∧ φ)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ⊆ wss 3257 × cxp 4770 ↾ cres 4774 {coprab 5527 |
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-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: resmpt2 5697 |
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