ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfiin3 GIF version

Theorem dfiin3 4769
Description: Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfiun3.1 𝐵 ∈ V
Assertion
Ref Expression
dfiin3 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Proof of Theorem dfiin3
StepHypRef Expression
1 dfiin3g 4767 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
2 dfiun3.1 . . 3 𝐵 ∈ V
32a1i 9 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 2466 1 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  Vcvv 2660   cint 3741   ciin 3784  cmpt 3959  ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-int 3742  df-iin 3786  df-br 3900  df-opab 3960  df-mpt 3961  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator