Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fsn2 | Unicode version |
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.) |
Ref | Expression |
---|---|
fsn2.1 |
Ref | Expression |
---|---|
fsn2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5242 | . . 3 | |
2 | fsn2.1 | . . . . 5 | |
3 | 2 | snid 3526 | . . . 4 |
4 | funfvex 5406 | . . . . 5 | |
5 | 4 | funfni 5193 | . . . 4 |
6 | 3, 5 | mpan2 421 | . . 3 |
7 | 1, 6 | syl 14 | . 2 |
8 | elex 2671 | . . 3 | |
9 | 8 | adantr 274 | . 2 |
10 | ffvelrn 5521 | . . . . . 6 | |
11 | 3, 10 | mpan2 421 | . . . . 5 |
12 | dffn3 5253 | . . . . . . . 8 | |
13 | 12 | biimpi 119 | . . . . . . 7 |
14 | imadmrn 4861 | . . . . . . . . . 10 | |
15 | fndm 5192 | . . . . . . . . . . 11 | |
16 | 15 | imaeq2d 4851 | . . . . . . . . . 10 |
17 | 14, 16 | syl5eqr 2164 | . . . . . . . . 9 |
18 | fnsnfv 5448 | . . . . . . . . . 10 | |
19 | 3, 18 | mpan2 421 | . . . . . . . . 9 |
20 | 17, 19 | eqtr4d 2153 | . . . . . . . 8 |
21 | feq3 5227 | . . . . . . . 8 | |
22 | 20, 21 | syl 14 | . . . . . . 7 |
23 | 13, 22 | mpbid 146 | . . . . . 6 |
24 | 1, 23 | syl 14 | . . . . 5 |
25 | 11, 24 | jca 304 | . . . 4 |
26 | snssi 3634 | . . . . 5 | |
27 | fss 5254 | . . . . . 6 | |
28 | 27 | ancoms 266 | . . . . 5 |
29 | 26, 28 | sylan 281 | . . . 4 |
30 | 25, 29 | impbii 125 | . . 3 |
31 | fsng 5561 | . . . . 5 | |
32 | 2, 31 | mpan 420 | . . . 4 |
33 | 32 | anbi2d 459 | . . 3 |
34 | 30, 33 | syl5bb 191 | . 2 |
35 | 7, 9, 34 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 wss 3041 csn 3497 cop 3500 cdm 4509 crn 4510 cima 4512 wfn 5088 wf 5089 cfv 5093 |
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-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 |
This theorem is referenced by: fnressn 5574 fressnfv 5575 mapsnconst 6556 elixpsn 6597 en1 6661 |
Copyright terms: Public domain | W3C validator |