Theorem inpreima 6342

Description: Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Assertion

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

Proof of Theorem inpreima

Step	Hyp	Ref	Expression
1		funcnvcnv 5956	. 2 |- ( Fun F -> Fun `' `' F )
2		imain 5974	. 2 |- ( Fun `' `' F -> ( `' F " ( A i^i B ) ) = ( ( `' F " A ) i^i ( `' F " B ) ) )
3	1, 2	syl 17	1 |- ( Fun F -> ( `' F " ( A i^i B ) ) = ( ( `' F " A ) i^i ( `' F " B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   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:  fimacnvinrn  6348  frnsuppeq  7307  ofco2  20257  cnrest2  21090  cnhaus  21158  kgencn3  21361  qtoptop2  21502  basqtop  21514  ismbfd  23407  mbfimaopn2  23424  i1fima  23445  i1fima2  23446  i1fd  23448  disjpreima  29397  sspreima  29447  smfco  41009