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Mirrors > Home > ILE Home > Th. List > setsfun0 | Unicode version |
Description: A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 11994 is useful for proofs based on isstruct2r 11970 which requires for to be an extensible structure. (Contributed by AV, 7-Jun-2021.) |
Ref | Expression |
---|---|
setsfun0 | sSet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 5164 | . . . . 5 | |
2 | 1 | ad2antlr 480 | . . . 4 |
3 | funsng 5169 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | dmres 4840 | . . . . . . 7 | |
6 | 5 | ineq1i 3273 | . . . . . 6 |
7 | in32 3288 | . . . . . . 7 | |
8 | incom 3268 | . . . . . . . . 9 | |
9 | disjdif 3435 | . . . . . . . . 9 | |
10 | 8, 9 | eqtri 2160 | . . . . . . . 8 |
11 | 10 | ineq1i 3273 | . . . . . . 7 |
12 | 0in 3398 | . . . . . . 7 | |
13 | 7, 11, 12 | 3eqtri 2164 | . . . . . 6 |
14 | 6, 13 | eqtri 2160 | . . . . 5 |
15 | 14 | a1i 9 | . . . 4 |
16 | funun 5167 | . . . 4 | |
17 | 2, 4, 15, 16 | syl21anc 1215 | . . 3 |
18 | difundir 3329 | . . . . 5 | |
19 | resdifcom 4837 | . . . . . . 7 | |
20 | 19 | a1i 9 | . . . . . 6 |
21 | elex 2697 | . . . . . . . . 9 | |
22 | elex 2697 | . . . . . . . . 9 | |
23 | opm 4156 | . . . . . . . . . 10 | |
24 | n0r 3376 | . . . . . . . . . 10 | |
25 | 23, 24 | sylbir 134 | . . . . . . . . 9 |
26 | 21, 22, 25 | syl2an 287 | . . . . . . . 8 |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | disjsn2 3586 | . . . . . . 7 | |
29 | disjdif2 3441 | . . . . . . 7 | |
30 | 27, 28, 29 | 3syl 17 | . . . . . 6 |
31 | 20, 30 | uneq12d 3231 | . . . . 5 |
32 | 18, 31 | syl5eq 2184 | . . . 4 |
33 | 32 | funeqd 5145 | . . 3 |
34 | 17, 33 | mpbird 166 | . 2 |
35 | simpll 518 | . . . . 5 | |
36 | opexg 4150 | . . . . . 6 | |
37 | 36 | adantl 275 | . . . . 5 |
38 | setsvalg 11989 | . . . . 5 sSet | |
39 | 35, 37, 38 | syl2anc 408 | . . . 4 sSet |
40 | 39 | difeq1d 3193 | . . 3 sSet |
41 | 40 | funeqd 5145 | . 2 sSet |
42 | 34, 41 | mpbird 166 | 1 sSet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wne 2308 cvv 2686 cdif 3068 cun 3069 cin 3070 c0 3363 csn 3527 cop 3530 cdm 4539 cres 4541 wfun 5117 (class class class)co 5774 sSet csts 11957 |
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-in1 603 ax-in2 604 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 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sets 11966 |
This theorem is referenced by: setsn0fun 11996 |
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