Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3538 | . . . 4 | |
2 | 1 | ixpeq1d 6604 | . . 3 |
3 | 2 | eleq2d 2209 | . 2 |
4 | opeq1 3705 | . . . . 5 | |
5 | 4 | sneqd 3540 | . . . 4 |
6 | 5 | eqeq2d 2151 | . . 3 |
7 | 6 | rexbidv 2438 | . 2 |
8 | elex 2697 | . . 3 | |
9 | vex 2689 | . . . . . . 7 | |
10 | vex 2689 | . . . . . . 7 | |
11 | 9, 10 | opex 4151 | . . . . . 6 |
12 | 11 | snex 4109 | . . . . 5 |
13 | eleq1 2202 | . . . . 5 | |
14 | 12, 13 | mpbiri 167 | . . . 4 |
15 | 14 | rexlimivw 2545 | . . 3 |
16 | eleq1 2202 | . . . 4 | |
17 | eqeq1 2146 | . . . . 5 | |
18 | 17 | rexbidv 2438 | . . . 4 |
19 | vex 2689 | . . . . . 6 | |
20 | 19 | elixp 6599 | . . . . 5 |
21 | fveq2 5421 | . . . . . . . 8 | |
22 | 21 | eleq1d 2208 | . . . . . . 7 |
23 | 9, 22 | ralsn 3567 | . . . . . 6 |
24 | 23 | anbi2i 452 | . . . . 5 |
25 | simpl 108 | . . . . . . . . 9 | |
26 | fveq2 5421 | . . . . . . . . . . . . 13 | |
27 | 26 | eleq1d 2208 | . . . . . . . . . . . 12 |
28 | 9, 27 | ralsn 3567 | . . . . . . . . . . 11 |
29 | 28 | biimpri 132 | . . . . . . . . . 10 |
30 | 29 | adantl 275 | . . . . . . . . 9 |
31 | ffnfv 5578 | . . . . . . . . 9 | |
32 | 25, 30, 31 | sylanbrc 413 | . . . . . . . 8 |
33 | 9 | fsn2 5594 | . . . . . . . 8 |
34 | 32, 33 | sylib 121 | . . . . . . 7 |
35 | opeq2 3706 | . . . . . . . . 9 | |
36 | 35 | sneqd 3540 | . . . . . . . 8 |
37 | 36 | rspceeqv 2807 | . . . . . . 7 |
38 | 34, 37 | syl 14 | . . . . . 6 |
39 | 9, 10 | fvsn 5615 | . . . . . . . . . 10 |
40 | id 19 | . . . . . . . . . 10 | |
41 | 39, 40 | eqeltrid 2226 | . . . . . . . . 9 |
42 | 9, 10 | fnsn 5177 | . . . . . . . . 9 |
43 | 41, 42 | jctil 310 | . . . . . . . 8 |
44 | fneq1 5211 | . . . . . . . . 9 | |
45 | fveq1 5420 | . . . . . . . . . 10 | |
46 | 45 | eleq1d 2208 | . . . . . . . . 9 |
47 | 44, 46 | anbi12d 464 | . . . . . . . 8 |
48 | 43, 47 | syl5ibrcom 156 | . . . . . . 7 |
49 | 48 | rexlimiv 2543 | . . . . . 6 |
50 | 38, 49 | impbii 125 | . . . . 5 |
51 | 20, 24, 50 | 3bitri 205 | . . . 4 |
52 | 16, 18, 51 | vtoclbg 2747 | . . 3 |
53 | 8, 15, 52 | pm5.21nii 693 | . 2 |
54 | 3, 7, 53 | vtoclbg 2747 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 cvv 2686 csn 3527 cop 3530 wfn 5118 wf 5119 cfv 5123 cixp 6592 |
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 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-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 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-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ixp 6593 |
This theorem is referenced by: ixpsnf1o 6630 |
Copyright terms: Public domain | W3C validator |