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Theorem riinrab 5002
 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 5000 . . 3 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝐴)
2 rzal 4455 . . . . 5 (𝑋 = ∅ → ∀𝑥𝑋 𝜑)
32ralrimivw 3187 . . . 4 (𝑋 = ∅ → ∀𝑦𝐴𝑥𝑋 𝜑)
4 rabid2 3386 . . . 4 (𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑} ↔ ∀𝑦𝐴𝑥𝑋 𝜑)
53, 4sylibr 235 . . 3 (𝑋 = ∅ → 𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
61, 5eqtrd 2860 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
7 ssrab2 4059 . . . . 5 {𝑦𝐴𝜑} ⊆ 𝐴
87rgenw 3154 . . . 4 𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴
9 riinn0 5001 . . . 4 ((∀𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
108, 9mpan 686 . . 3 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
11 iinrab 4987 . . 3 (𝑋 ≠ ∅ → 𝑥𝑋 {𝑦𝐴𝜑} = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
1210, 11eqtrd 2860 . 2 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
136, 12pm2.61ine 3104 1 (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   ≠ wne 3020  ∀wral 3142  {crab 3146   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ∩ ciin 4917 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295  df-iin 4919 This theorem is referenced by:  acsfn1  16924  acsfn1c  16925  acsfn2  16926  cntziinsn  18397  acsfn1p  19500  csscld  23767
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