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Theorem inrab2 4275
Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
Assertion
Ref Expression
inrab2 ({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem inrab2
StepHypRef Expression
1 df-rab 3147 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abid1 2956 . . 3 𝐵 = {𝑥𝑥𝐵}
31, 2ineq12i 4186 . 2 ({𝑥𝐴𝜑} ∩ 𝐵) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵})
4 df-rab 3147 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)}
5 inab 4270 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ 𝑥𝐵)}
6 elin 4168 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
76anbi1i 625 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
8 an32 644 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ 𝑥𝐵))
97, 8bitri 277 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ 𝑥𝐵))
109abbii 2886 . . . 4 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ 𝑥𝐵)}
115, 10eqtr4i 2847 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵}) = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)}
124, 11eqtr4i 2847 . 2 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵})
133, 12eqtr4i 2847 1 ({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  wcel 2110  {cab 2799  {crab 3142  cin 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942
This theorem is referenced by:  iooval2  12770  fzval2  12894  smuval2  15830  smueqlem  15838  dfphi2  16110  ordtrest  21809  ordtrest2lem  21810  ordtrestNEW  31164  ordtrest2NEWlem  31165  itg2addnclem2  34943  dmatALTbas  44455
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