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Mirrors > Home > ILE Home > Th. List > funbrfv | Unicode version |
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
funbrfv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 5233 |
. . . 4
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2 | brrelex2 4667 |
. . . 4
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3 | 1, 2 | sylan 283 |
. . 3
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4 | breq2 4007 |
. . . . . 6
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5 | 4 | anbi2d 464 |
. . . . 5
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6 | eqeq2 2187 |
. . . . 5
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7 | 5, 6 | imbi12d 234 |
. . . 4
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8 | funeu 5241 |
. . . . . 6
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9 | tz6.12-1 5542 |
. . . . . 6
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10 | 8, 9 | sylan2 286 |
. . . . 5
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11 | 10 | anabss7 583 |
. . . 4
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12 | 7, 11 | vtoclg 2797 |
. . 3
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13 | 3, 12 | mpcom 36 |
. 2
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14 | 13 | ex 115 |
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-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
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-sbc 2963 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-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 |
This theorem is referenced by: funopfv 5555 fnbrfvb 5556 fvelima 5567 fvi 5573 fmptco 5682 fliftfun 5796 fliftval 5800 tfrlem5 6314 sum0 11391 isumz 11392 fsumsersdc 11398 isumclim 11424 zprodap0 11584 dvaddxx 14060 dvmulxx 14061 dvcj 14066 dvrecap 14070 dvef 14081 pilem3 14097 |
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