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Theorem iinrab 5074
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 4507 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)))
21abbidv 2806 . 2 (𝐴 ≠ ∅ → {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)})
3 df-rab 3434 . . . . 5 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
43a1i 11 . . . 4 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
54iineq2i 5019 . . 3 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
6 iinab 5073 . . 3 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
75, 6eqtri 2763 . 2 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
8 df-rab 3434 . 2 {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)}
92, 7, 83eqtr4g 2800 1 (𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  {crab 3433  c0 4339   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-nul 4340  df-iin 4999
This theorem is referenced by:  iinrab2  5075  riinrab  5089  ubthlem1  30899  pmapglbx  39752  preimageiingt  46676  preimaleiinlt  46677
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