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Theorem riinrab 4532
 Description: Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinrab (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem riinrab
StepHypRef Expression
1 riin0 4530 . . 3 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝐴)
2 rzal 4025 . . . . 5 (𝑋 = ∅ → ∀𝑥𝑋 𝜑)
32ralrimivw 2950 . . . 4 (𝑋 = ∅ → ∀𝑦𝐴𝑥𝑋 𝜑)
4 rabid2 3096 . . . 4 (𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑} ↔ ∀𝑦𝐴𝑥𝑋 𝜑)
53, 4sylibr 223 . . 3 (𝑋 = ∅ → 𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
61, 5eqtrd 2644 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
7 ssrab2 3650 . . . . 5 {𝑦𝐴𝜑} ⊆ 𝐴
87rgenw 2908 . . . 4 𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴
9 riinn0 4531 . . . 4 ((∀𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
108, 9mpan 702 . . 3 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
11 iinrab 4518 . . 3 (𝑋 ≠ ∅ → 𝑥𝑋 {𝑦𝐴𝜑} = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
1210, 11eqtrd 2644 . 2 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
136, 12pm2.61ine 2865 1 (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ≠ wne 2780  ∀wral 2896  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-iin 4458 This theorem is referenced by:  acsfn1  16145  acsfn1c  16146  acsfn2  16147  cntziinsn  17590  csscld  22856  acsfn1p  36788
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