Theorem inpreima 5666

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 5297	. 2 |- ( Fun F -> Fun `' `' F )
2		imain 5320	. 2 |- ( Fun `' `' F -> ( `' F " ( A i^i B ) ) = ( ( `' F " A ) i^i ( `' F " B ) ) )
3	1, 2	syl 14	1 |- ( Fun F -> ( `' F " ( A i^i B ) ) = ( ( `' F " A ) i^i ( `' F " B ) ) )

Syntax hints:   -> wi 4   = wceq 1364   i^i cin 3143   `'ccnv 4646   "cima 4650   Fun wfun 5232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-fun 5240

This theorem is referenced by:  nn0supp  9263  cnrest2  14221