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Theorem imain 4924
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 "  i^i  F "  i^i  F "

Proof of Theorem imain
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imainlem 4923 . . 3  F
"  i^i  C_  F "  i^i  F "
21a1i 9 . 2  Fun  `' F  F "  i^i  C_  F "  i^i  F "
3 eeanv 1804 . . . . . 6  F  F  F  F
4 simprll 489 . . . . . . . . . . 11  Fun  `' F  F  F
5 simpr 103 . . . . . . . . . . . . . 14  F  F
6 simpr 103 . . . . . . . . . . . . . 14  F  F
75, 6anim12i 321 . . . . . . . . . . . . 13  F  F  F  F
8 funcnveq 4905 . . . . . . . . . . . . . . . . 17  Fun  `' F  F  F
98biimpi 113 . . . . . . . . . . . . . . . 16  Fun  `' F  F  F
10919.21bi 1447 . . . . . . . . . . . . . . 15  Fun  `' F  F  F
111019.21bbi 1448 . . . . . . . . . . . . . 14  Fun  `' F  F  F
1211imp 115 . . . . . . . . . . . . 13  Fun  `' F  F  F
137, 12sylan2 270 . . . . . . . . . . . 12  Fun  `' F  F  F
14 simprrl 491 . . . . . . . . . . . 12  Fun  `' F  F  F
1513, 14eqeltrd 2111 . . . . . . . . . . 11  Fun  `' F  F  F
16 elin 3120 . . . . . . . . . . 11  i^i
174, 15, 16sylanbrc 394 . . . . . . . . . 10  Fun  `' F  F  F  i^i
18 simprlr 490 . . . . . . . . . 10  Fun  `' F  F  F  F
1917, 18jca 290 . . . . . . . . 9  Fun  `' F  F  F  i^i  F
2019ex 108 . . . . . . . 8  Fun  `' F  F  F  i^i  F
2120exlimdv 1697 . . . . . . 7  Fun  `' F  F  F  i^i  F
2221eximdv 1757 . . . . . 6  Fun  `' F  F  F  i^i  F
233, 22syl5bir 142 . . . . 5  Fun  `' F  F  F  i^i  F
24 df-rex 2306 . . . . . 6  F  F
25 df-rex 2306 . . . . . 6  F  F
2624, 25anbi12i 433 . . . . 5  F  F  F  F
27 df-rex 2306 . . . . 5  i^i  F  i^i  F
2823, 26, 273imtr4g 194 . . . 4  Fun  `' F  F  F  i^i  F
2928ss2abdv 3007 . . 3  Fun  `' F  {  |  F  F }  C_ 
{  |  i^i  F }
30 dfima2 4613 . . . . 5  F
"  {  |  F }
31 dfima2 4613 . . . . 5  F
"  {  |  F }
3230, 31ineq12i 3130 . . . 4  F "  i^i  F "  {  |  F }  i^i  {  |  F }
33 inab 3199 . . . 4  {  |  F }  i^i  {  |  F }  {  |  F  F }
3432, 33eqtri 2057 . . 3  F "  i^i  F "  {  |  F  F }
35 dfima2 4613 . . 3  F
"  i^i  {  |  i^i  F }
3629, 34, 353sstr4g 2980 . 2  Fun  `' F  F "  i^i  F "  C_  F "  i^i
372, 36eqssd 2956 1  Fun  `' F  F "  i^i  F "  i^i  F "
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242  wex 1378   wcel 1390   {cab 2023  wrex 2301    i^i cin 2910    C_ wss 2911   class class class wbr 3755   `'ccnv 4287   "cima 4291   Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by:  inpreima  5236
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