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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 3063 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | df-rab 3063 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
3 | 1, 2 | ineq12i 4111 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
4 | df-rab 3063 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} | |
5 | inab 4199 | . . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓))} | |
6 | anandi 676 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
7 | 6 | abbii 2804 | . . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓))} |
8 | 5, 7 | eqtr4i 2765 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} |
9 | 4, 8 | eqtr4i 2765 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
10 | 3, 9 | eqtr4i 2765 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2114 {cab 2717 {crab 3058 ∩ cin 3852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-in 3860 |
This theorem is referenced by: rabnc 4286 ixxin 12851 hashbclem 13915 phiprmpw 16226 submacs 18120 ablfacrp 19320 dfrhm2 19604 ordtbaslem 21952 ordtbas2 21955 ordtopn3 21960 ordtcld3 21963 ordthauslem 22147 pthaus 22402 xkohaus 22417 tsmsfbas 22892 minveclem3b 24193 shftmbl 24303 mumul 25931 ppiub 25953 lgsquadlem2 26130 umgrislfupgrlem 27080 numedglnl 27102 clwwlknondisj 28061 frcond3 28219 numclwwlk3lem2 28334 xppreima 30570 xpinpreima 31441 xpinpreima2 31442 measvuni 31765 subfacp1lem6 32731 satfv1 32909 cnambfre 35481 itg2addnclem2 35485 ftc1anclem6 35511 refsymrels2 36335 dfeqvrels2 36357 refrelsredund4 36401 anrabdioph 40215 undisjrab 41503 smfaddlem2 43879 smfmullem4 43908 |
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