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Mirrors > Home > ILE Home > Th. List > fmptpr | Unicode version |
Description: Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Ref | Expression |
---|---|
fmptpr.1 | |
fmptpr.2 | |
fmptpr.3 | |
fmptpr.4 | |
fmptpr.5 | |
fmptpr.6 |
Ref | Expression |
---|---|
fmptpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3534 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | mpt0 5250 | . . . . . 6 | |
4 | 3 | uneq1i 3226 | . . . . 5 |
5 | uncom 3220 | . . . . 5 | |
6 | un0 3396 | . . . . 5 | |
7 | 4, 5, 6 | 3eqtri 2164 | . . . 4 |
8 | fmptpr.1 | . . . . . 6 | |
9 | elex 2697 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | fmptpr.3 | . . . . . 6 | |
12 | elex 2697 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | uncom 3220 | . . . . . . 7 | |
15 | un0 3396 | . . . . . . 7 | |
16 | 14, 15 | eqtr3i 2162 | . . . . . 6 |
17 | 16 | a1i 9 | . . . . 5 |
18 | fmptpr.5 | . . . . 5 | |
19 | 10, 13, 17, 18 | fmptapd 5611 | . . . 4 |
20 | 7, 19 | syl5eqr 2186 | . . 3 |
21 | 20 | uneq1d 3229 | . 2 |
22 | fmptpr.2 | . . . 4 | |
23 | elex 2697 | . . . 4 | |
24 | 22, 23 | syl 14 | . . 3 |
25 | fmptpr.4 | . . . 4 | |
26 | elex 2697 | . . . 4 | |
27 | 25, 26 | syl 14 | . . 3 |
28 | df-pr 3534 | . . . . 5 | |
29 | 28 | eqcomi 2143 | . . . 4 |
30 | 29 | a1i 9 | . . 3 |
31 | fmptpr.6 | . . 3 | |
32 | 24, 27, 30, 31 | fmptapd 5611 | . 2 |
33 | 2, 21, 32 | 3eqtrd 2176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cun 3069 c0 3363 csn 3527 cpr 3528 cop 3530 cmpt 3989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: (None) |
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