Theorem inpreima 5781

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 5396	. 2 |- ( Fun F -> Fun `' `' F )
2		imain 5419	. 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 1398   i^i cin 3200   `'ccnv 4730   "cima 4734   Fun wfun 5327

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335

This theorem is referenced by:  fsuppeq  6425  fsuppeqg  6426  nn0supp  9515  cnrest2  15047