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Mirrors > Home > ILE Home > Th. List > fvmptss2 | Unicode version |
Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fvmptss2.1 | |
fvmptss2.2 |
Ref | Expression |
---|---|
fvmptss2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvss 5403 | . 2 | |
2 | fvmptss2.2 | . . . . . 6 | |
3 | 2 | funmpt2 5132 | . . . . 5 |
4 | funrel 5110 | . . . . 5 | |
5 | 3, 4 | ax-mp 5 | . . . 4 |
6 | 5 | brrelex1i 4552 | . . 3 |
7 | nfcv 2258 | . . . 4 | |
8 | nfmpt1 3991 | . . . . . . 7 | |
9 | 2, 8 | nfcxfr 2255 | . . . . . 6 |
10 | nfcv 2258 | . . . . . 6 | |
11 | 7, 9, 10 | nfbr 3944 | . . . . 5 |
12 | nfv 1493 | . . . . 5 | |
13 | 11, 12 | nfim 1536 | . . . 4 |
14 | breq1 3902 | . . . . 5 | |
15 | fvmptss2.1 | . . . . . 6 | |
16 | 15 | sseq2d 3097 | . . . . 5 |
17 | 14, 16 | imbi12d 233 | . . . 4 |
18 | df-br 3900 | . . . . 5 | |
19 | opabid 4149 | . . . . . . 7 | |
20 | eqimss 3121 | . . . . . . . 8 | |
21 | 20 | adantl 275 | . . . . . . 7 |
22 | 19, 21 | sylbi 120 | . . . . . 6 |
23 | df-mpt 3961 | . . . . . . 7 | |
24 | 2, 23 | eqtri 2138 | . . . . . 6 |
25 | 22, 24 | eleq2s 2212 | . . . . 5 |
26 | 18, 25 | sylbi 120 | . . . 4 |
27 | 7, 13, 17, 26 | vtoclgf 2718 | . . 3 |
28 | 6, 27 | mpcom 36 | . 2 |
29 | 1, 28 | mpg 1412 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cvv 2660 wss 3041 cop 3500 class class class wbr 3899 copab 3958 cmpt 3959 wrel 4514 wfun 5087 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: mptfvex 5474 |
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