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Theorem riinrab 4728
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 4726 . . 3 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝐴)
2 rzal 4212 . . . . 5 (𝑋 = ∅ → ∀𝑥𝑋 𝜑)
32ralrimivw 3115 . . . 4 (𝑋 = ∅ → ∀𝑦𝐴𝑥𝑋 𝜑)
4 rabid2 3266 . . . 4 (𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑} ↔ ∀𝑦𝐴𝑥𝑋 𝜑)
53, 4sylibr 224 . . 3 (𝑋 = ∅ → 𝐴 = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
61, 5eqtrd 2804 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
7 ssrab2 3834 . . . . 5 {𝑦𝐴𝜑} ⊆ 𝐴
87rgenw 3072 . . . 4 𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴
9 riinn0 4727 . . . 4 ((∀𝑥𝑋 {𝑦𝐴𝜑} ⊆ 𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
108, 9mpan 662 . . 3 (𝑋 ≠ ∅ → (𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = 𝑥𝑋 {𝑦𝐴𝜑})
11 iinrab 4714 . . 3 (𝑋 ≠ ∅ → 𝑥𝑋 {𝑦𝐴𝜑} = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑})
1210, 11eqtrd 2804 . 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 1630  wne 2942  wral 3060  {crab 3064  cin 3720  wss 3721  c0 4061   ciin 4653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-in 3728  df-ss 3735  df-nul 4062  df-iin 4655
This theorem is referenced by:  acsfn1  16528  acsfn1c  16529  acsfn2  16530  cntziinsn  17973  csscld  23266  acsfn1p  38288
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