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| Mirrors > Home > ILE Home > Th. List > fvmptf | Unicode version | ||
| Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5654 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) |
| Ref | Expression |
|---|---|
| fvmptf.1 |
|
| fvmptf.2 |
|
| fvmptf.3 |
|
| fvmptf.4 |
|
| Ref | Expression |
|---|---|
| fvmptf |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 2782 |
. . 3
| |
| 2 | fvmptf.1 |
. . . 4
| |
| 3 | fvmptf.2 |
. . . . . 6
| |
| 4 | 3 | nfel1 2358 |
. . . . 5
|
| 5 | fvmptf.4 |
. . . . . . . 8
| |
| 6 | nfmpt1 4136 |
. . . . . . . 8
| |
| 7 | 5, 6 | nfcxfr 2344 |
. . . . . . 7
|
| 8 | 7, 2 | nffv 5585 |
. . . . . 6
|
| 9 | 8, 3 | nfeq 2355 |
. . . . 5
|
| 10 | 4, 9 | nfim 1594 |
. . . 4
|
| 11 | fvmptf.3 |
. . . . . 6
| |
| 12 | 11 | eleq1d 2273 |
. . . . 5
|
| 13 | fveq2 5575 |
. . . . . 6
| |
| 14 | 13, 11 | eqeq12d 2219 |
. . . . 5
|
| 15 | 12, 14 | imbi12d 234 |
. . . 4
|
| 16 | 5 | fvmpt2 5662 |
. . . . 5
|
| 17 | 16 | ex 115 |
. . . 4
|
| 18 | 2, 10, 15, 17 | vtoclgaf 2837 |
. . 3
|
| 19 | 1, 18 | syl5 32 |
. 2
|
| 20 | 19 | imp 124 |
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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-sbc 2998 df-csb 3093 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 |
| This theorem is referenced by: fvmptd3 5672 elfvmptrab1 5673 sumrbdclem 11659 fsum3 11669 isumss 11673 prodrbdclem 11853 prodmodclem2a 11858 zproddc 11861 fprodntrivap 11866 prodssdc 11871 pcmpt 12637 |
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