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Mirrors > Home > ILE Home > Th. List > fvmptssdm | Unicode version |
Description: If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.) |
Ref | Expression |
---|---|
fvmpt2.1 |
Ref | Expression |
---|---|
fvmptssdm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5421 | . . . . . 6 | |
2 | 1 | sseq1d 3126 | . . . . 5 |
3 | 2 | imbi2d 229 | . . . 4 |
4 | nfrab1 2610 | . . . . . . 7 | |
5 | 4 | nfcri 2275 | . . . . . 6 |
6 | nfra1 2466 | . . . . . . 7 | |
7 | fvmpt2.1 | . . . . . . . . . 10 | |
8 | nfmpt1 4021 | . . . . . . . . . 10 | |
9 | 7, 8 | nfcxfr 2278 | . . . . . . . . 9 |
10 | nfcv 2281 | . . . . . . . . 9 | |
11 | 9, 10 | nffv 5431 | . . . . . . . 8 |
12 | nfcv 2281 | . . . . . . . 8 | |
13 | 11, 12 | nfss 3090 | . . . . . . 7 |
14 | 6, 13 | nfim 1551 | . . . . . 6 |
15 | 5, 14 | nfim 1551 | . . . . 5 |
16 | eleq1 2202 | . . . . . 6 | |
17 | fveq2 5421 | . . . . . . . 8 | |
18 | 17 | sseq1d 3126 | . . . . . . 7 |
19 | 18 | imbi2d 229 | . . . . . 6 |
20 | 16, 19 | imbi12d 233 | . . . . 5 |
21 | 7 | dmmpt 5034 | . . . . . . 7 |
22 | 21 | eleq2i 2206 | . . . . . 6 |
23 | 21 | rabeq2i 2683 | . . . . . . . . . 10 |
24 | 7 | fvmpt2 5504 | . . . . . . . . . . 11 |
25 | eqimss 3151 | . . . . . . . . . . 11 | |
26 | 24, 25 | syl 14 | . . . . . . . . . 10 |
27 | 23, 26 | sylbi 120 | . . . . . . . . 9 |
28 | 27 | adantr 274 | . . . . . . . 8 |
29 | 7 | dmmptss 5035 | . . . . . . . . . 10 |
30 | 29 | sseli 3093 | . . . . . . . . 9 |
31 | rsp 2480 | . . . . . . . . 9 | |
32 | 30, 31 | mpan9 279 | . . . . . . . 8 |
33 | 28, 32 | sstrd 3107 | . . . . . . 7 |
34 | 33 | ex 114 | . . . . . 6 |
35 | 22, 34 | sylbir 134 | . . . . 5 |
36 | 15, 20, 35 | chvar 1730 | . . . 4 |
37 | 3, 36 | vtoclga 2752 | . . 3 |
38 | 37, 21 | eleq2s 2234 | . 2 |
39 | 38 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 crab 2420 cvv 2686 wss 3071 cmpt 3989 cdm 4539 cfv 5123 |
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-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 |
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-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 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-fv 5131 |
This theorem is referenced by: (None) |
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