Theorem inpreima 5293

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 4958	. 2 |- ( Fun F -> Fun `' `' F )
2		imain 4981	. 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 1243   i^i cin 2916   `'ccnv 4344   "cima 4348   Fun wfun 4896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904

This theorem is referenced by:  nn0supp  8234