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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 5465 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 2671 | . . 3 | |
2 | fvmptf.1 | . . . 4 | |
3 | fvmptf.2 | . . . . . 6 | |
4 | 3 | nfel1 2269 | . . . . 5 |
5 | fvmptf.4 | . . . . . . . 8 | |
6 | nfmpt1 3991 | . . . . . . . 8 | |
7 | 5, 6 | nfcxfr 2255 | . . . . . . 7 |
8 | 7, 2 | nffv 5399 | . . . . . 6 |
9 | 8, 3 | nfeq 2266 | . . . . 5 |
10 | 4, 9 | nfim 1536 | . . . 4 |
11 | fvmptf.3 | . . . . . 6 | |
12 | 11 | eleq1d 2186 | . . . . 5 |
13 | fveq2 5389 | . . . . . 6 | |
14 | 13, 11 | eqeq12d 2132 | . . . . 5 |
15 | 12, 14 | imbi12d 233 | . . . 4 |
16 | 5 | fvmpt2 5472 | . . . . 5 |
17 | 16 | ex 114 | . . . 4 |
18 | 2, 10, 15, 17 | vtoclgaf 2725 | . . 3 |
19 | 1, 18 | syl5 32 | . 2 |
20 | 19 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wnfc 2245 cvv 2660 cmpt 3959 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-sbc 2883 df-csb 2976 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-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: fvmptd3 5482 elfvmptrab1 5483 sumrbdclem 11100 fsum3 11111 isumss 11115 |
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