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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 5066 | . . 3 | |
2 | fsn2.1 | . . . . 5 | |
3 | 2 | snid 3425 | . . . 4 |
4 | funfvex 5212 | . . . . 5 | |
5 | 4 | funfni 5019 | . . . 4 |
6 | 3, 5 | mpan2 415 | . . 3 |
7 | 1, 6 | syl 14 | . 2 |
8 | elex 2610 | . . 3 | |
9 | 8 | adantr 270 | . 2 |
10 | ffvelrn 5321 | . . . . . 6 | |
11 | 3, 10 | mpan2 415 | . . . . 5 |
12 | dffn3 5073 | . . . . . . . 8 | |
13 | 12 | biimpi 118 | . . . . . . 7 |
14 | imadmrn 4698 | . . . . . . . . . 10 | |
15 | fndm 5018 | . . . . . . . . . . 11 | |
16 | 15 | imaeq2d 4688 | . . . . . . . . . 10 |
17 | 14, 16 | syl5eqr 2127 | . . . . . . . . 9 |
18 | fnsnfv 5253 | . . . . . . . . . 10 | |
19 | 3, 18 | mpan2 415 | . . . . . . . . 9 |
20 | 17, 19 | eqtr4d 2116 | . . . . . . . 8 |
21 | feq3 5052 | . . . . . . . 8 | |
22 | 20, 21 | syl 14 | . . . . . . 7 |
23 | 13, 22 | mpbid 145 | . . . . . 6 |
24 | 1, 23 | syl 14 | . . . . 5 |
25 | 11, 24 | jca 300 | . . . 4 |
26 | snssi 3529 | . . . . 5 | |
27 | fss 5074 | . . . . . 6 | |
28 | 27 | ancoms 264 | . . . . 5 |
29 | 26, 28 | sylan 277 | . . . 4 |
30 | 25, 29 | impbii 124 | . . 3 |
31 | fsng 5357 | . . . . 5 | |
32 | 2, 31 | mpan 414 | . . . 4 |
33 | 32 | anbi2d 451 | . . 3 |
34 | 30, 33 | syl5bb 190 | . 2 |
35 | 7, 9, 34 | pm5.21nii 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wcel 1433 cvv 2601 wss 2973 csn 3398 cop 3401 cdm 4363 crn 4364 cima 4366 wfn 4917 wf 4918 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: fnressn 5370 fressnfv 5371 en1 6302 |
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