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Theorem imainss 4682
Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
imainss  R "  i^i  C_  R "  i^i  `' R "

Proof of Theorem imainss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . . . . 11 
_V
2 vex 2554 . . . . . . . . . . 11 
_V
31, 2brcnv 4461 . . . . . . . . . 10  `' R  R
4 19.8a 1479 . . . . . . . . . 10  `' R  `' R
53, 4sylan2br 272 . . . . . . . . 9  R  `' R
65ancoms 255 . . . . . . . 8  R  `' R
76anim2i 324 . . . . . . 7  R  `' R
8 simprl 483 . . . . . . 7  R  R
97, 8jca 290 . . . . . 6  R  `' R  R
109anassrs 380 . . . . 5  R  `' R  R
11 elin 3120 . . . . . . 7  i^i  `' R "  `' R "
122elima2 4617 . . . . . . . 8  `' R "  `' R
1312anbi2i 430 . . . . . . 7  `' R "  `' R
1411, 13bitri 173 . . . . . 6  i^i  `' R "  `' R
1514anbi1i 431 . . . . 5  i^i  `' R "  R  `' R  R
1610, 15sylibr 137 . . . 4  R  i^i  `' R "  R
1716eximi 1488 . . 3  R  i^i  `' R "  R
181elima2 4617 . . . . 5  R "  R
1918anbi1i 431 . . . 4  R
"  R
20 elin 3120 . . . 4  R
"  i^i  R "
21 19.41v 1779 . . . 4  R  R
2219, 20, 213bitr4i 201 . . 3  R
"  i^i  R
231elima2 4617 . . 3  R "  i^i  `' R "  i^i  `' R "  R
2417, 22, 233imtr4i 190 . 2  R
"  i^i  R "  i^i  `' R "
2524ssriv 2943 1  R "  i^i  C_  R "  i^i  `' R "
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1378   wcel 1390    i^i cin 2910    C_ wss 2911   class class class wbr 3755   `'ccnv 4287   "cima 4291
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-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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