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Mirrors > Home > MPE Home > Th. List > restval | Structured version Visualization version Unicode version |
Description: The subspace topology
induced by the topology ![]() ![]() |
Ref | Expression |
---|---|
restval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3053 |
. 2
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2 | elex 3053 |
. 2
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3 | mptexg 6133 |
. . . . 5
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4 | rnexg 6722 |
. . . . 5
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5 | 3, 4 | syl 17 |
. . . 4
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6 | 5 | adantr 467 |
. . 3
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7 | simpl 459 |
. . . . . 6
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8 | simpr 463 |
. . . . . . 7
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9 | 8 | ineq2d 3633 |
. . . . . 6
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10 | 7, 9 | mpteq12dv 4480 |
. . . . 5
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11 | 10 | rneqd 5061 |
. . . 4
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12 | df-rest 15314 |
. . . 4
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13 | 11, 12 | ovmpt2ga 6423 |
. . 3
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14 | 6, 13 | mpd3an3 1364 |
. 2
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15 | 1, 2, 14 | syl2an 480 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-rest 15314 |
This theorem is referenced by: elrest 15319 0rest 15321 restid2 15322 tgrest 20168 resttopon 20170 restco 20173 rest0 20178 restfpw 20188 neitr 20189 ordtrest2 20213 1stcrest 20461 2ndcrest 20462 kgencmp 20553 xkoptsub 20662 trfilss 20897 trfg 20899 uzrest 20905 restmetu 21578 ellimc2 22825 limcflf 22829 ordtrest2NEW 28722 ptrest 31932 |
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