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Theorem iinrab 4614
Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinrab (𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
Distinct variable groups:   𝑦,𝐴,𝑥   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem iinrab
StepHypRef Expression
1 r19.28zv 4099 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)))
21abbidv 2770 . 2 (𝐴 ≠ ∅ → {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)})
3 df-rab 2950 . . . . 5 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
43a1i 11 . . . 4 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
54iineq2i 4572 . . 3 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
6 iinab 4613 . . 3 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
75, 6eqtri 2673 . 2 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
8 df-rab 2950 . 2 {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)}
92, 7, 83eqtr4g 2710 1 (𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  {crab 2945  c0 3948   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-nul 3949  df-iin 4555
This theorem is referenced by:  iinrab2  4615  riinrab  4628  ubthlem1  27854  pmapglbx  35373  preimageiingt  41251  preimaleiinlt  41252
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