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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 3508 | . . . 4 | |
2 | 1 | ixpeq1d 6572 | . . 3 |
3 | 2 | eleq2d 2187 | . 2 |
4 | opeq1 3675 | . . . . 5 | |
5 | 4 | sneqd 3510 | . . . 4 |
6 | 5 | eqeq2d 2129 | . . 3 |
7 | 6 | rexbidv 2415 | . 2 |
8 | elex 2671 | . . 3 | |
9 | vex 2663 | . . . . . . 7 | |
10 | vex 2663 | . . . . . . 7 | |
11 | 9, 10 | opex 4121 | . . . . . 6 |
12 | 11 | snex 4079 | . . . . 5 |
13 | eleq1 2180 | . . . . 5 | |
14 | 12, 13 | mpbiri 167 | . . . 4 |
15 | 14 | rexlimivw 2522 | . . 3 |
16 | eleq1 2180 | . . . 4 | |
17 | eqeq1 2124 | . . . . 5 | |
18 | 17 | rexbidv 2415 | . . . 4 |
19 | vex 2663 | . . . . . 6 | |
20 | 19 | elixp 6567 | . . . . 5 |
21 | fveq2 5389 | . . . . . . . 8 | |
22 | 21 | eleq1d 2186 | . . . . . . 7 |
23 | 9, 22 | ralsn 3537 | . . . . . 6 |
24 | 23 | anbi2i 452 | . . . . 5 |
25 | simpl 108 | . . . . . . . . 9 | |
26 | fveq2 5389 | . . . . . . . . . . . . 13 | |
27 | 26 | eleq1d 2186 | . . . . . . . . . . . 12 |
28 | 9, 27 | ralsn 3537 | . . . . . . . . . . 11 |
29 | 28 | biimpri 132 | . . . . . . . . . 10 |
30 | 29 | adantl 275 | . . . . . . . . 9 |
31 | ffnfv 5546 | . . . . . . . . 9 | |
32 | 25, 30, 31 | sylanbrc 413 | . . . . . . . 8 |
33 | 9 | fsn2 5562 | . . . . . . . 8 |
34 | 32, 33 | sylib 121 | . . . . . . 7 |
35 | opeq2 3676 | . . . . . . . . 9 | |
36 | 35 | sneqd 3510 | . . . . . . . 8 |
37 | 36 | rspceeqv 2781 | . . . . . . 7 |
38 | 34, 37 | syl 14 | . . . . . 6 |
39 | 9, 10 | fvsn 5583 | . . . . . . . . . 10 |
40 | id 19 | . . . . . . . . . 10 | |
41 | 39, 40 | eqeltrid 2204 | . . . . . . . . 9 |
42 | 9, 10 | fnsn 5147 | . . . . . . . . 9 |
43 | 41, 42 | jctil 310 | . . . . . . . 8 |
44 | fneq1 5181 | . . . . . . . . 9 | |
45 | fveq1 5388 | . . . . . . . . . 10 | |
46 | 45 | eleq1d 2186 | . . . . . . . . 9 |
47 | 44, 46 | anbi12d 464 | . . . . . . . 8 |
48 | 43, 47 | syl5ibrcom 156 | . . . . . . 7 |
49 | 48 | rexlimiv 2520 | . . . . . 6 |
50 | 38, 49 | impbii 125 | . . . . 5 |
51 | 20, 24, 50 | 3bitri 205 | . . . 4 |
52 | 16, 18, 51 | vtoclbg 2721 | . . 3 |
53 | 8, 15, 52 | pm5.21nii 678 | . 2 |
54 | 3, 7, 53 | vtoclbg 2721 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 wrex 2394 cvv 2660 csn 3497 cop 3500 wfn 5088 wf 5089 cfv 5093 cixp 6560 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ixp 6561 |
This theorem is referenced by: ixpsnf1o 6598 |
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