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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 5469 | . . . . . 6 | |
2 | 1 | sseq1d 3157 | . . . . 5 |
3 | 2 | imbi2d 229 | . . . 4 |
4 | nfrab1 2636 | . . . . . . 7 | |
5 | 4 | nfcri 2293 | . . . . . 6 |
6 | nfra1 2488 | . . . . . . 7 | |
7 | fvmpt2.1 | . . . . . . . . . 10 | |
8 | nfmpt1 4058 | . . . . . . . . . 10 | |
9 | 7, 8 | nfcxfr 2296 | . . . . . . . . 9 |
10 | nfcv 2299 | . . . . . . . . 9 | |
11 | 9, 10 | nffv 5479 | . . . . . . . 8 |
12 | nfcv 2299 | . . . . . . . 8 | |
13 | 11, 12 | nfss 3121 | . . . . . . 7 |
14 | 6, 13 | nfim 1552 | . . . . . 6 |
15 | 5, 14 | nfim 1552 | . . . . 5 |
16 | eleq1 2220 | . . . . . 6 | |
17 | fveq2 5469 | . . . . . . . 8 | |
18 | 17 | sseq1d 3157 | . . . . . . 7 |
19 | 18 | imbi2d 229 | . . . . . 6 |
20 | 16, 19 | imbi12d 233 | . . . . 5 |
21 | 7 | dmmpt 5082 | . . . . . . 7 |
22 | 21 | eleq2i 2224 | . . . . . 6 |
23 | 21 | rabeq2i 2709 | . . . . . . . . . 10 |
24 | 7 | fvmpt2 5552 | . . . . . . . . . . 11 |
25 | eqimss 3182 | . . . . . . . . . . 11 | |
26 | 24, 25 | syl 14 | . . . . . . . . . 10 |
27 | 23, 26 | sylbi 120 | . . . . . . . . 9 |
28 | 27 | adantr 274 | . . . . . . . 8 |
29 | 7 | dmmptss 5083 | . . . . . . . . . 10 |
30 | 29 | sseli 3124 | . . . . . . . . 9 |
31 | rsp 2504 | . . . . . . . . 9 | |
32 | 30, 31 | mpan9 279 | . . . . . . . 8 |
33 | 28, 32 | sstrd 3138 | . . . . . . 7 |
34 | 33 | ex 114 | . . . . . 6 |
35 | 22, 34 | sylbir 134 | . . . . 5 |
36 | 15, 20, 35 | chvar 1737 | . . . 4 |
37 | 3, 36 | vtoclga 2778 | . . 3 |
38 | 37, 21 | eleq2s 2252 | . 2 |
39 | 38 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 crab 2439 cvv 2712 wss 3102 cmpt 4026 cdm 4587 cfv 5171 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fv 5179 |
This theorem is referenced by: (None) |
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