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Mirrors > Home > ILE Home > Th. List > fmpt | Unicode version |
Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmpt.1 |
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Ref | Expression |
---|---|
fmpt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpt.1 |
. . . 4
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2 | 1 | fnmpt 5174 |
. . 3
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3 | 1 | rnmpt 4715 |
. . . 4
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4 | r19.29 2520 |
. . . . . . 7
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5 | eleq1 2157 |
. . . . . . . . 9
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6 | 5 | biimparc 294 |
. . . . . . . 8
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7 | 6 | rexlimivw 2498 |
. . . . . . 7
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8 | 4, 7 | syl 14 |
. . . . . 6
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9 | 8 | ex 114 |
. . . . 5
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10 | 9 | abssdv 3110 |
. . . 4
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11 | 3, 10 | syl5eqss 3085 |
. . 3
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12 | df-f 5053 |
. . 3
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13 | 2, 11, 12 | sylanbrc 409 |
. 2
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14 | 1 | mptpreima 4958 |
. . . 4
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15 | fimacnv 5467 |
. . . 4
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16 | 14, 15 | syl5reqr 2142 |
. . 3
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17 | rabid2 2557 |
. . 3
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18 | 16, 17 | sylib 121 |
. 2
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19 | 13, 18 | impbii 125 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 |
This theorem is referenced by: f1ompt 5489 fmpti 5490 fmptd 5491 fmptdf 5494 rnmptss 5498 f1oresrab 5502 idref 5574 f1mpt 5588 f1stres 5968 f2ndres 5969 fmpt2x 6008 fmpt2co 6019 iunon 6087 mptelixpg 6531 dom2lem 6569 uzf 9121 cnmpt11 12105 cncfmpt1f 12348 mulcncflem 12353 mulcncf 12354 |
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