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Mirrors > Home > MPE Home > Th. List > inrab | Structured version Visualization version GIF version |
Description: Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.) |
Ref | Expression |
---|---|
inrab | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3072 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | df-rab 3072 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
3 | 1, 2 | ineq12i 4141 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
4 | df-rab 3072 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} | |
5 | inab 4230 | . . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓))} | |
6 | anandi 672 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
7 | 6 | abbii 2809 | . . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓))} |
8 | 5, 7 | eqtr4i 2769 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} |
9 | 4, 8 | eqtr4i 2769 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
10 | 3, 9 | eqtr4i 2769 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2108 {cab 2715 {crab 3067 ∩ cin 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1542 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3072 df-v 3424 df-in 3890 |
This theorem is referenced by: rabnc 4318 ixxin 13025 hashbclem 14092 phiprmpw 16405 submacs 18380 ablfacrp 19584 dfrhm2 19876 ordtbaslem 22247 ordtbas2 22250 ordtopn3 22255 ordtcld3 22258 ordthauslem 22442 pthaus 22697 xkohaus 22712 tsmsfbas 23187 minveclem3b 24497 shftmbl 24607 mumul 26235 ppiub 26257 lgsquadlem2 26434 umgrislfupgrlem 27395 numedglnl 27417 clwwlknondisj 28376 frcond3 28534 numclwwlk3lem2 28649 xppreima 30884 xpinpreima 31758 xpinpreima2 31759 measvuni 32082 subfacp1lem6 33047 satfv1 33225 cnambfre 35752 itg2addnclem2 35756 ftc1anclem6 35782 refsymrels2 36606 dfeqvrels2 36628 refrelsredund4 36672 anrabdioph 40518 undisjrab 41813 smfaddlem2 44186 smfmullem4 44215 |
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