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Mirrors > Home > ILE Home > Th. List > elixpsn | Unicode version |
Description: Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
elixpsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3543 | . . . 4 | |
2 | 1 | ixpeq1d 6612 | . . 3 |
3 | 2 | eleq2d 2210 | . 2 |
4 | opeq1 3713 | . . . . 5 | |
5 | 4 | sneqd 3545 | . . . 4 |
6 | 5 | eqeq2d 2152 | . . 3 |
7 | 6 | rexbidv 2439 | . 2 |
8 | elex 2700 | . . 3 | |
9 | vex 2692 | . . . . . . 7 | |
10 | vex 2692 | . . . . . . 7 | |
11 | 9, 10 | opex 4159 | . . . . . 6 |
12 | 11 | snex 4117 | . . . . 5 |
13 | eleq1 2203 | . . . . 5 | |
14 | 12, 13 | mpbiri 167 | . . . 4 |
15 | 14 | rexlimivw 2548 | . . 3 |
16 | eleq1 2203 | . . . 4 | |
17 | eqeq1 2147 | . . . . 5 | |
18 | 17 | rexbidv 2439 | . . . 4 |
19 | vex 2692 | . . . . . 6 | |
20 | 19 | elixp 6607 | . . . . 5 |
21 | fveq2 5429 | . . . . . . . 8 | |
22 | 21 | eleq1d 2209 | . . . . . . 7 |
23 | 9, 22 | ralsn 3574 | . . . . . 6 |
24 | 23 | anbi2i 453 | . . . . 5 |
25 | simpl 108 | . . . . . . . . 9 | |
26 | fveq2 5429 | . . . . . . . . . . . . 13 | |
27 | 26 | eleq1d 2209 | . . . . . . . . . . . 12 |
28 | 9, 27 | ralsn 3574 | . . . . . . . . . . 11 |
29 | 28 | biimpri 132 | . . . . . . . . . 10 |
30 | 29 | adantl 275 | . . . . . . . . 9 |
31 | ffnfv 5586 | . . . . . . . . 9 | |
32 | 25, 30, 31 | sylanbrc 414 | . . . . . . . 8 |
33 | 9 | fsn2 5602 | . . . . . . . 8 |
34 | 32, 33 | sylib 121 | . . . . . . 7 |
35 | opeq2 3714 | . . . . . . . . 9 | |
36 | 35 | sneqd 3545 | . . . . . . . 8 |
37 | 36 | rspceeqv 2811 | . . . . . . 7 |
38 | 34, 37 | syl 14 | . . . . . 6 |
39 | 9, 10 | fvsn 5623 | . . . . . . . . . 10 |
40 | id 19 | . . . . . . . . . 10 | |
41 | 39, 40 | eqeltrid 2227 | . . . . . . . . 9 |
42 | 9, 10 | fnsn 5185 | . . . . . . . . 9 |
43 | 41, 42 | jctil 310 | . . . . . . . 8 |
44 | fneq1 5219 | . . . . . . . . 9 | |
45 | fveq1 5428 | . . . . . . . . . 10 | |
46 | 45 | eleq1d 2209 | . . . . . . . . 9 |
47 | 44, 46 | anbi12d 465 | . . . . . . . 8 |
48 | 43, 47 | syl5ibrcom 156 | . . . . . . 7 |
49 | 48 | rexlimiv 2546 | . . . . . 6 |
50 | 38, 49 | impbii 125 | . . . . 5 |
51 | 20, 24, 50 | 3bitri 205 | . . . 4 |
52 | 16, 18, 51 | vtoclbg 2750 | . . 3 |
53 | 8, 15, 52 | pm5.21nii 694 | . 2 |
54 | 3, 7, 53 | vtoclbg 2750 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1332 wcel 1481 wral 2417 wrex 2418 cvv 2689 csn 3532 cop 3535 wfn 5126 wf 5127 cfv 5131 cixp 6600 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ixp 6601 |
This theorem is referenced by: ixpsnf1o 6638 |
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