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Mirrors > Home > ILE Home > Th. List > mpt2xopovel | Unicode version |
Description: Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpt2xopoveq.f |
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Ref | Expression |
---|---|
mpt2xopovel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2xopoveq.f |
. . . 4
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2 | 1 | mpt2xopn0yelv 6004 |
. . 3
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3 | 2 | pm4.71rd 386 |
. 2
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4 | 1 | mpt2xopoveq 6005 |
. . . . . 6
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5 | 4 | eleq2d 2157 |
. . . . 5
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6 | nfcv 2228 |
. . . . . . 7
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7 | 6 | elrabsf 2877 |
. . . . . 6
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8 | sbccom 2914 |
. . . . . . . 8
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9 | sbccom 2914 |
. . . . . . . . 9
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10 | 9 | sbcbii 2898 |
. . . . . . . 8
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11 | 8, 10 | bitri 182 |
. . . . . . 7
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12 | 11 | anbi2i 445 |
. . . . . 6
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13 | 7, 12 | bitri 182 |
. . . . 5
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14 | 5, 13 | syl6bb 194 |
. . . 4
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15 | 14 | pm5.32da 440 |
. . 3
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16 | 3anass 928 |
. . 3
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17 | 15, 16 | syl6bbr 196 |
. 2
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18 | 3, 17 | bitrd 186 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 |
This theorem is referenced by: (None) |
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