Theorem intima0 37939

Description: Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intima0	|- |^|_ a e. A ( a " B ) = |^| { x | E. a e. A x = ( a " B ) }

Distinct variable groups:   x, A   x, B   x, a

Allowed substitution hints:   A( a)   B( a)

Proof of Theorem intima0

Step	Hyp	Ref	Expression
1		vex 3203	. . 3 |- a e. _V
2		imaexg 7103	. . 3 |- ( a e. _V -> ( a " B ) e. _V )
3	1, 2	ax-mp 5	. 2 |- ( a " B ) e. _V
4	3	dfiin2 4555	1 |- |^|_ a e. A ( a " B ) = |^| { x | E. a e. A x = ( a " B ) }

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   |^|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-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:  intimass2  37947  intimasn2  37950