| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5590 |
. 2
| |
| 2 | fvmptss2.2 |
. . . . . 6
| |
| 3 | 2 | funmpt2 5310 |
. . . . 5
|
| 4 | funrel 5288 |
. . . . 5
| |
| 5 | 3, 4 | ax-mp 5 |
. . . 4
|
| 6 | 5 | brrelex1i 4718 |
. . 3
|
| 7 | nfcv 2348 |
. . . 4
| |
| 8 | nfmpt1 4137 |
. . . . . . 7
| |
| 9 | 2, 8 | nfcxfr 2345 |
. . . . . 6
|
| 10 | nfcv 2348 |
. . . . . 6
| |
| 11 | 7, 9, 10 | nfbr 4090 |
. . . . 5
|
| 12 | nfv 1551 |
. . . . 5
| |
| 13 | 11, 12 | nfim 1595 |
. . . 4
|
| 14 | breq1 4047 |
. . . . 5
| |
| 15 | fvmptss2.1 |
. . . . . 6
| |
| 16 | 15 | sseq2d 3223 |
. . . . 5
|
| 17 | 14, 16 | imbi12d 234 |
. . . 4
|
| 18 | df-br 4045 |
. . . . 5
| |
| 19 | opabid 4302 |
. . . . . . 7
| |
| 20 | eqimss 3247 |
. . . . . . . 8
| |
| 21 | 20 | adantl 277 |
. . . . . . 7
|
| 22 | 19, 21 | sylbi 121 |
. . . . . 6
|
| 23 | df-mpt 4107 |
. . . . . . 7
| |
| 24 | 2, 23 | eqtri 2226 |
. . . . . 6
|
| 25 | 22, 24 | eleq2s 2300 |
. . . . 5
|
| 26 | 18, 25 | sylbi 121 |
. . . 4
|
| 27 | 7, 13, 17, 26 | vtoclgf 2831 |
. . 3
|
| 28 | 6, 27 | mpcom 36 |
. 2
|
| 29 | 1, 28 | mpg 1474 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-iota 5232 df-fun 5273 df-fv 5279 |
| This theorem is referenced by: mptfvex 5665 |
| Copyright terms: Public domain | W3C validator |