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Theorem iinrab 4977
Description: Indexed intersection of a restricted class abstraction. (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 4429 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)))
21abbidv 2888 . 2 (𝐴 ≠ ∅ → {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)})
3 df-rab 3142 . . . . 5 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
43a1i 11 . . . 4 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
54iineq2i 4927 . . 3 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
6 iinab 4976 . . 3 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
75, 6eqtri 2847 . 2 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
8 df-rab 3142 . 2 {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)}
92, 7, 83eqtr4g 2884 1 (𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wral 3133  {crab 3137  c0 4276   ciin 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-nul 4277  df-iin 4908
This theorem is referenced by:  iinrab2  4978  riinrab  4992  ubthlem1  28660  pmapglbx  37014  preimageiingt  43286  preimaleiinlt  43287
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