Theorem intimasn2 37950

Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intimasn2	|- ( B e. V -> ( |^| A " { B } ) = |^|_ x e. A ( x " { B } ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem intimasn2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intimasn 37949	. 2 |- ( B e. V -> ( |^| A " { B } ) = |^| { y | E. x e. A y = ( x " { B } ) } )
2		intima0 37939	. 2 |- |^|_ x e. A ( x " { B } ) = |^| { y | E. x e. A y = ( x " { B } ) }
3	1, 2	syl6eqr 2674	1 |- ( B e. V -> ( |^| A " { B } ) = |^|_ x e. A ( x " { B } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {csn 4177   |^|cint 4475   |^|_ciin 4521   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iin 4523  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)