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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 3588 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | mpt0 5323 | . . . . . 6 | |
4 | 3 | uneq1i 3277 | . . . . 5 |
5 | uncom 3271 | . . . . 5 | |
6 | un0 3447 | . . . . 5 | |
7 | 4, 5, 6 | 3eqtri 2195 | . . . 4 |
8 | fmptpr.1 | . . . . . 6 | |
9 | elex 2741 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | fmptpr.3 | . . . . . 6 | |
12 | elex 2741 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | uncom 3271 | . . . . . . 7 | |
15 | un0 3447 | . . . . . . 7 | |
16 | 14, 15 | eqtr3i 2193 | . . . . . 6 |
17 | 16 | a1i 9 | . . . . 5 |
18 | fmptpr.5 | . . . . 5 | |
19 | 10, 13, 17, 18 | fmptapd 5685 | . . . 4 |
20 | 7, 19 | eqtr3id 2217 | . . 3 |
21 | 20 | uneq1d 3280 | . 2 |
22 | fmptpr.2 | . . . 4 | |
23 | elex 2741 | . . . 4 | |
24 | 22, 23 | syl 14 | . . 3 |
25 | fmptpr.4 | . . . 4 | |
26 | elex 2741 | . . . 4 | |
27 | 25, 26 | syl 14 | . . 3 |
28 | df-pr 3588 | . . . . 5 | |
29 | 28 | eqcomi 2174 | . . . 4 |
30 | 29 | a1i 9 | . . 3 |
31 | fmptpr.6 | . . 3 | |
32 | 24, 27, 30, 31 | fmptapd 5685 | . 2 |
33 | 2, 21, 32 | 3eqtrd 2207 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cvv 2730 cun 3119 c0 3414 csn 3581 cpr 3582 cop 3584 cmpt 4048 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 |
This theorem is referenced by: (None) |
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