Theorem imain 5974

Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

Assertion

Ref	Expression
imain	|- ( Fun `' F -> ( F " ( A i^i B ) ) = ( ( F " A ) i^i ( F " B ) ) )

Proof of Theorem imain

Step	Hyp	Ref	Expression
1		imadif 5973	. . 3 |- ( Fun `' F -> ( F " ( A \ ( A \ B ) ) ) = ( ( F " A ) \ ( F " ( A \ B ) ) ) )
2		imadif 5973	. . . 4 |- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
3	2	difeq2d 3728	. . 3 |- ( Fun `' F -> ( ( F " A ) \ ( F " ( A \ B ) ) ) = ( ( F " A ) \ ( ( F " A ) \ ( F " B ) ) ) )
4	1, 3	eqtrd 2656	. 2 |- ( Fun `' F -> ( F " ( A \ ( A \ B ) ) ) = ( ( F " A ) \ ( ( F " A ) \ ( F " B ) ) ) )
5		dfin4 3867	. . 3 |- ( A i^i B ) = ( A \ ( A \ B ) )
6	5	imaeq2i 5464	. 2 |- ( F " ( A i^i B ) ) = ( F " ( A \ ( A \ B ) ) )
7		dfin4 3867	. 2 |- ( ( F " A ) i^i ( F " B ) ) = ( ( F " A ) \ ( ( F " A ) \ ( F " B ) ) )
8	4, 6, 7	3eqtr4g 2681	1 |- ( Fun `' F -> ( F " ( A i^i B ) ) = ( ( F " A ) i^i ( F " B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   \ cdif 3571   i^i cin 3573   `'ccnv 5113   "cima 5117   Fun wfun 5882

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-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

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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  inpreima  6342  rnelfmlem  21756  fmfnfmlem3  21760  spthispth  26622  ballotlemfrc  30588  poimirlem1  33410  poimirlem2  33411  poimirlem3  33412  poimirlem4  33413  poimirlem6  33415  poimirlem7  33416  poimirlem11  33420  poimirlem12  33421  poimirlem16  33425  poimirlem17  33426  poimirlem19  33428  poimirlem20  33429  poimirlem23  33432  poimirlem24  33433  poimirlem25  33434  poimirlem29  33438  poimirlem31  33440