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Mirrors > Home > ILE Home > Th. List > resiexg | Unicode version |
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) |
Ref | Expression |
---|---|
resiexg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4935 |
. . 3
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2 | simpr 110 |
. . . . 5
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3 | eleq1 2240 |
. . . . . 6
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4 | 3 | biimpa 296 |
. . . . 5
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5 | 2, 4 | jca 306 |
. . . 4
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6 | vex 2740 |
. . . . . 6
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7 | 6 | opelres 4912 |
. . . . 5
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8 | df-br 4004 |
. . . . . . 7
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9 | 6 | ideq 4779 |
. . . . . . 7
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10 | 8, 9 | bitr3i 186 |
. . . . . 6
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11 | 10 | anbi1i 458 |
. . . . 5
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12 | 7, 11 | bitri 184 |
. . . 4
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13 | opelxp 4656 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 5, 12, 13 | 3imtr4i 201 |
. . 3
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15 | 1, 14 | relssi 4717 |
. 2
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16 | xpexg 4740 |
. . 3
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17 | 16 | anidms 397 |
. 2
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18 | ssexg 4142 |
. 2
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19 | 15, 17, 18 | sylancr 414 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-res 4638 |
This theorem is referenced by: ordiso 7034 omct 7115 ctssexmid 7147 ssomct 12440 ndxarg 12479 subctctexmid 14670 |
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