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Theorem inopab 4391
Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
inopab ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ψ)}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem inopab
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4387 . . 3 Rel {⟨x, y⟩ ∣ φ}
2 relin1 4378 . . 3 (Rel {⟨x, y⟩ ∣ φ} → Rel ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}))
31, 2ax-mp 7 . 2 Rel ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ})
4 relopab 4387 . 2 Rel {⟨x, y⟩ ∣ (φ ψ)}
5 sban 1807 . . . 4 ([w / y]([z / x]φ [z / x]ψ) ↔ ([w / y][z / x]φ [w / y][z / x]ψ))
6 sban 1807 . . . . 5 ([z / x](φ ψ) ↔ ([z / x]φ [z / x]ψ))
76sbbii 1626 . . . 4 ([w / y][z / x](φ ψ) ↔ [w / y]([z / x]φ [z / x]ψ))
8 opelopabsbALT 3966 . . . . 5 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ [w / y][z / x]φ)
9 opelopabsbALT 3966 . . . . 5 (⟨z, w {⟨x, y⟩ ∣ ψ} ↔ [w / y][z / x]ψ)
108, 9anbi12i 436 . . . 4 ((⟨z, w {⟨x, y⟩ ∣ φ} z, w {⟨x, y⟩ ∣ ψ}) ↔ ([w / y][z / x]φ [w / y][z / x]ψ))
115, 7, 103bitr4ri 202 . . 3 ((⟨z, w {⟨x, y⟩ ∣ φ} z, w {⟨x, y⟩ ∣ ψ}) ↔ [w / y][z / x](φ ψ))
12 elin 3099 . . 3 (⟨z, w ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ (⟨z, w {⟨x, y⟩ ∣ φ} z, w {⟨x, y⟩ ∣ ψ}))
13 opelopabsbALT 3966 . . 3 (⟨z, w {⟨x, y⟩ ∣ (φ ψ)} ↔ [w / y][z / x](φ ψ))
1411, 12, 133bitr4i 201 . 2 (⟨z, w ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ ⟨z, w {⟨x, y⟩ ∣ (φ ψ)})
153, 4, 14eqrelriiv 4357 1 ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ψ)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226   wcel 1370  [wsb 1623  cin 2889  cop 3349  {copab 3787  Rel wrel 4273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789  df-xp 4274  df-rel 4275
This theorem is referenced by:  inxp  4393  resopab  4575  cnvin  4654  fndmin  5195  enq0enq  6280
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