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Theorem riinrab 5084
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 5082 . . 3 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝐴)
2 rzal 4509 . . . . 5 (𝑋 = ∅ → ∀𝑥𝑋 𝜑)
32ralrimivw 3150 . . . 4 (𝑋 = ∅ → ∀𝑦𝐴𝑥𝑋 𝜑)
4 rabid2 3470 . . . 4 (𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑} ↔ ∀𝑦𝐴𝑥𝑋 𝜑)
53, 4sylibr 234 . . 3 (𝑋 = ∅ → 𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
61, 5eqtrd 2777 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
7 ssrab2 4080 . . . . 5 {𝑦𝐴𝜑} ⊆ 𝐴
87rgenw 3065 . . . 4 𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴
9 riinn0 5083 . . . 4 ((∀𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
108, 9mpan 690 . . 3 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
11 iinrab 5069 . . 3 (𝑋 ≠ ∅ → 𝑥𝑋 {𝑦𝐴𝜑} = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
1210, 11eqtrd 2777 . 2 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
136, 12pm2.61ine 3025 1 (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wne 2940  wral 3061  {crab 3436  cin 3950  wss 3951  c0 4333   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-iin 4994
This theorem is referenced by:  acsfn1  17704  acsfn1c  17705  acsfn2  17706  cntziinsn  19355  acsfn1p  20800  csscld  25283
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