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Mirrors > Home > ILE Home > Th. List > mapsncnv | Unicode version |
Description: Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | |
mapsncnv.b | |
mapsncnv.x | |
mapsncnv.f |
Ref | Expression |
---|---|
mapsncnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 6627 | . . . . . . . . 9 | |
2 | mapsncnv.x | . . . . . . . . . 10 | |
3 | 2 | snid 3601 | . . . . . . . . 9 |
4 | ffvelrn 5612 | . . . . . . . . 9 | |
5 | 1, 3, 4 | sylancl 410 | . . . . . . . 8 |
6 | eqid 2164 | . . . . . . . . 9 | |
7 | mapsncnv.b | . . . . . . . . 9 | |
8 | 6, 7, 2 | mapsnconst 6651 | . . . . . . . 8 |
9 | 5, 8 | jca 304 | . . . . . . 7 |
10 | eleq1 2227 | . . . . . . . 8 | |
11 | sneq 3581 | . . . . . . . . . 10 | |
12 | 11 | xpeq2d 4622 | . . . . . . . . 9 |
13 | 12 | eqeq2d 2176 | . . . . . . . 8 |
14 | 10, 13 | anbi12d 465 | . . . . . . 7 |
15 | 9, 14 | syl5ibrcom 156 | . . . . . 6 |
16 | 15 | imp 123 | . . . . 5 |
17 | fconst6g 5380 | . . . . . . . . 9 | |
18 | 2 | snex 4158 | . . . . . . . . . 10 |
19 | 7, 18 | elmap 6634 | . . . . . . . . 9 |
20 | 17, 19 | sylibr 133 | . . . . . . . 8 |
21 | vex 2724 | . . . . . . . . . . 11 | |
22 | 21 | fvconst2 5695 | . . . . . . . . . 10 |
23 | 3, 22 | mp1i 10 | . . . . . . . . 9 |
24 | 23 | eqcomd 2170 | . . . . . . . 8 |
25 | 20, 24 | jca 304 | . . . . . . 7 |
26 | eleq1 2227 | . . . . . . . 8 | |
27 | fveq1 5479 | . . . . . . . . 9 | |
28 | 27 | eqeq2d 2176 | . . . . . . . 8 |
29 | 26, 28 | anbi12d 465 | . . . . . . 7 |
30 | 25, 29 | syl5ibrcom 156 | . . . . . 6 |
31 | 30 | imp 123 | . . . . 5 |
32 | 16, 31 | impbii 125 | . . . 4 |
33 | mapsncnv.s | . . . . . . 7 | |
34 | 33 | oveq2i 5847 | . . . . . 6 |
35 | 34 | eleq2i 2231 | . . . . 5 |
36 | 35 | anbi1i 454 | . . . 4 |
37 | 33 | xpeq1i 4618 | . . . . . 6 |
38 | 37 | eqeq2i 2175 | . . . . 5 |
39 | 38 | anbi2i 453 | . . . 4 |
40 | 32, 36, 39 | 3bitr4i 211 | . . 3 |
41 | 40 | opabbii 4043 | . 2 |
42 | mapsncnv.f | . . . . 5 | |
43 | df-mpt 4039 | . . . . 5 | |
44 | 42, 43 | eqtri 2185 | . . . 4 |
45 | 44 | cnveqi 4773 | . . 3 |
46 | cnvopab 4999 | . . 3 | |
47 | 45, 46 | eqtri 2185 | . 2 |
48 | df-mpt 4039 | . 2 | |
49 | 41, 47, 48 | 3eqtr4i 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wcel 2135 cvv 2721 csn 3570 copab 4036 cmpt 4037 cxp 4596 ccnv 4597 wf 5178 cfv 5182 (class class class)co 5836 cmap 6605 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-map 6607 |
This theorem is referenced by: mapsnf1o2 6653 mapsnf1o3 6654 |
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