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Theorem riinrab 5081
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 5079 . . 3 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝐴)
2 rzal 4504 . . . . 5 (𝑋 = ∅ → ∀𝑥𝑋 𝜑)
32ralrimivw 3145 . . . 4 (𝑋 = ∅ → ∀𝑦𝐴𝑥𝑋 𝜑)
4 rabid2 3459 . . . 4 (𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑} ↔ ∀𝑦𝐴𝑥𝑋 𝜑)
53, 4sylibr 233 . . 3 (𝑋 = ∅ → 𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
61, 5eqtrd 2767 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
7 ssrab2 4073 . . . . 5 {𝑦𝐴𝜑} ⊆ 𝐴
87rgenw 3060 . . . 4 𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴
9 riinn0 5080 . . . 4 ((∀𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
108, 9mpan 689 . . 3 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
11 iinrab 5066 . . 3 (𝑋 ≠ ∅ → 𝑥𝑋 {𝑦𝐴𝜑} = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
1210, 11eqtrd 2767 . 2 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
136, 12pm2.61ine 3020 1 (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wne 2935  wral 3056  {crab 3427  cin 3943  wss 3944  c0 4318   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-iin 4994
This theorem is referenced by:  acsfn1  17632  acsfn1c  17633  acsfn2  17634  cntziinsn  19279  acsfn1p  20676  csscld  25164
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