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Theorem dfiin3g 5672
Description: Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfiin3g (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))

Proof of Theorem dfiin3g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfiin2g 4821 . 2 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 eqid 2772 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5664 . . 3 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
43inteqi 4747 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
51, 4syl6eqr 2826 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  {cab 2752  wral 3082  wrex 3083   cint 4743   ciin 4787  cmpt 5002  ran crn 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-int 4744  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-cnv 5409  df-dm 5411  df-rn 5412
This theorem is referenced by:  dfiin3  5674  riinint  5675  iinon  7775  cmpfi  21714
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