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Mirrors > Home > ILE Home > Th. List > ixpsnf1o | Unicode version |
Description: A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
ixpsnf1o.f |
Ref | Expression |
---|---|
ixpsnf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpsnf1o.f | . 2 | |
2 | snexg 4108 | . . . 4 | |
3 | vex 2689 | . . . . 5 | |
4 | 3 | snex 4109 | . . . 4 |
5 | xpexg 4653 | . . . 4 | |
6 | 2, 4, 5 | sylancl 409 | . . 3 |
7 | 6 | adantr 274 | . 2 |
8 | vex 2689 | . . . . 5 | |
9 | 8 | rnex 4806 | . . . 4 |
10 | 9 | uniex 4359 | . . 3 |
11 | 10 | a1i 9 | . 2 |
12 | sneq 3538 | . . . . . 6 | |
13 | 12 | xpeq1d 4562 | . . . . 5 |
14 | 13 | eqeq2d 2151 | . . . 4 |
15 | 14 | anbi2d 459 | . . 3 |
16 | elixpsn 6629 | . . . . . 6 | |
17 | 16 | elv 2690 | . . . . 5 |
18 | 12 | ixpeq1d 6604 | . . . . . 6 |
19 | 18 | eleq2d 2209 | . . . . 5 |
20 | 17, 19 | syl5bbr 193 | . . . 4 |
21 | 20 | anbi1d 460 | . . 3 |
22 | vex 2689 | . . . . . . 7 | |
23 | 22, 3 | xpsn 5596 | . . . . . 6 |
24 | 23 | eqeq2i 2150 | . . . . 5 |
25 | 24 | anbi2i 452 | . . . 4 |
26 | eqid 2139 | . . . . . . . . 9 | |
27 | opeq2 3706 | . . . . . . . . . . 11 | |
28 | 27 | sneqd 3540 | . . . . . . . . . 10 |
29 | 28 | rspceeqv 2807 | . . . . . . . . 9 |
30 | 26, 29 | mpan2 421 | . . . . . . . 8 |
31 | 22, 3 | op2nda 5023 | . . . . . . . . 9 |
32 | 31 | eqcomi 2143 | . . . . . . . 8 |
33 | 30, 32 | jctir 311 | . . . . . . 7 |
34 | eqeq1 2146 | . . . . . . . . 9 | |
35 | 34 | rexbidv 2438 | . . . . . . . 8 |
36 | rneq 4766 | . . . . . . . . . 10 | |
37 | 36 | unieqd 3747 | . . . . . . . . 9 |
38 | 37 | eqeq2d 2151 | . . . . . . . 8 |
39 | 35, 38 | anbi12d 464 | . . . . . . 7 |
40 | 33, 39 | syl5ibrcom 156 | . . . . . 6 |
41 | 40 | imp 123 | . . . . 5 |
42 | vex 2689 | . . . . . . . . . . 11 | |
43 | 22, 42 | op2nda 5023 | . . . . . . . . . 10 |
44 | 43 | eqeq2i 2150 | . . . . . . . . 9 |
45 | eqidd 2140 | . . . . . . . . . . 11 | |
46 | 45 | ancli 321 | . . . . . . . . . 10 |
47 | eleq1w 2200 | . . . . . . . . . . 11 | |
48 | opeq2 3706 | . . . . . . . . . . . . 13 | |
49 | 48 | sneqd 3540 | . . . . . . . . . . . 12 |
50 | 49 | eqeq2d 2151 | . . . . . . . . . . 11 |
51 | 47, 50 | anbi12d 464 | . . . . . . . . . 10 |
52 | 46, 51 | syl5ibrcom 156 | . . . . . . . . 9 |
53 | 44, 52 | syl5bi 151 | . . . . . . . 8 |
54 | rneq 4766 | . . . . . . . . . . 11 | |
55 | 54 | unieqd 3747 | . . . . . . . . . 10 |
56 | 55 | eqeq2d 2151 | . . . . . . . . 9 |
57 | eqeq1 2146 | . . . . . . . . . 10 | |
58 | 57 | anbi2d 459 | . . . . . . . . 9 |
59 | 56, 58 | imbi12d 233 | . . . . . . . 8 |
60 | 53, 59 | syl5ibrcom 156 | . . . . . . 7 |
61 | 60 | rexlimiv 2543 | . . . . . 6 |
62 | 61 | imp 123 | . . . . 5 |
63 | 41, 62 | impbii 125 | . . . 4 |
64 | 25, 63 | bitri 183 | . . 3 |
65 | 15, 21, 64 | vtoclbg 2747 | . 2 |
66 | 1, 7, 11, 65 | f1od 5973 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 cvv 2686 csn 3527 cop 3530 cuni 3736 cmpt 3989 cxp 4537 crn 4540 wf1o 5122 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-13 1491 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 ax-un 4355 |
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: mapsnf1o 6631 |
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