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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 5496 | . . . . . 6 | |
2 | 1 | sseq1d 3176 | . . . . 5 |
3 | 2 | imbi2d 229 | . . . 4 |
4 | nfrab1 2649 | . . . . . . 7 | |
5 | 4 | nfcri 2306 | . . . . . 6 |
6 | nfra1 2501 | . . . . . . 7 | |
7 | fvmpt2.1 | . . . . . . . . . 10 | |
8 | nfmpt1 4082 | . . . . . . . . . 10 | |
9 | 7, 8 | nfcxfr 2309 | . . . . . . . . 9 |
10 | nfcv 2312 | . . . . . . . . 9 | |
11 | 9, 10 | nffv 5506 | . . . . . . . 8 |
12 | nfcv 2312 | . . . . . . . 8 | |
13 | 11, 12 | nfss 3140 | . . . . . . 7 |
14 | 6, 13 | nfim 1565 | . . . . . 6 |
15 | 5, 14 | nfim 1565 | . . . . 5 |
16 | eleq1 2233 | . . . . . 6 | |
17 | fveq2 5496 | . . . . . . . 8 | |
18 | 17 | sseq1d 3176 | . . . . . . 7 |
19 | 18 | imbi2d 229 | . . . . . 6 |
20 | 16, 19 | imbi12d 233 | . . . . 5 |
21 | 7 | dmmpt 5106 | . . . . . . 7 |
22 | 21 | eleq2i 2237 | . . . . . 6 |
23 | 21 | rabeq2i 2727 | . . . . . . . . . 10 |
24 | 7 | fvmpt2 5579 | . . . . . . . . . . 11 |
25 | eqimss 3201 | . . . . . . . . . . 11 | |
26 | 24, 25 | syl 14 | . . . . . . . . . 10 |
27 | 23, 26 | sylbi 120 | . . . . . . . . 9 |
28 | 27 | adantr 274 | . . . . . . . 8 |
29 | 7 | dmmptss 5107 | . . . . . . . . . 10 |
30 | 29 | sseli 3143 | . . . . . . . . 9 |
31 | rsp 2517 | . . . . . . . . 9 | |
32 | 30, 31 | mpan9 279 | . . . . . . . 8 |
33 | 28, 32 | sstrd 3157 | . . . . . . 7 |
34 | 33 | ex 114 | . . . . . 6 |
35 | 22, 34 | sylbir 134 | . . . . 5 |
36 | 15, 20, 35 | chvar 1750 | . . . 4 |
37 | 3, 36 | vtoclga 2796 | . . 3 |
38 | 37, 21 | eleq2s 2265 | . 2 |
39 | 38 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 crab 2452 cvv 2730 wss 3121 cmpt 4050 cdm 4611 cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fv 5206 |
This theorem is referenced by: (None) |
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