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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 5331 | . . 3 | |
2 | fsn2.1 | . . . . 5 | |
3 | 2 | snid 3601 | . . . 4 |
4 | funfvex 5497 | . . . . 5 | |
5 | 4 | funfni 5282 | . . . 4 |
6 | 3, 5 | mpan2 422 | . . 3 |
7 | 1, 6 | syl 14 | . 2 |
8 | elex 2732 | . . 3 | |
9 | 8 | adantr 274 | . 2 |
10 | ffvelrn 5612 | . . . . . 6 | |
11 | 3, 10 | mpan2 422 | . . . . 5 |
12 | dffn3 5342 | . . . . . . . 8 | |
13 | 12 | biimpi 119 | . . . . . . 7 |
14 | imadmrn 4950 | . . . . . . . . . 10 | |
15 | fndm 5281 | . . . . . . . . . . 11 | |
16 | 15 | imaeq2d 4940 | . . . . . . . . . 10 |
17 | 14, 16 | eqtr3id 2211 | . . . . . . . . 9 |
18 | fnsnfv 5539 | . . . . . . . . . 10 | |
19 | 3, 18 | mpan2 422 | . . . . . . . . 9 |
20 | 17, 19 | eqtr4d 2200 | . . . . . . . 8 |
21 | feq3 5316 | . . . . . . . 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 3711 | . . . . 5 | |
27 | fss 5343 | . . . . . 6 | |
28 | 27 | ancoms 266 | . . . . 5 |
29 | 26, 28 | sylan 281 | . . . 4 |
30 | 25, 29 | impbii 125 | . . 3 |
31 | fsng 5652 | . . . . 5 | |
32 | 2, 31 | mpan 421 | . . . 4 |
33 | 32 | anbi2d 460 | . . 3 |
34 | 30, 33 | syl5bb 191 | . 2 |
35 | 7, 9, 34 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 wss 3111 csn 3570 cop 3573 cdm 4598 crn 4599 cima 4601 wfn 5177 wf 5178 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 |
This theorem is referenced by: fnressn 5665 fressnfv 5666 mapsnconst 6651 elixpsn 6692 en1 6756 |
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