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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 5544 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 2723 | . . 3 | |
2 | fvmptf.1 | . . . 4 | |
3 | fvmptf.2 | . . . . . 6 | |
4 | 3 | nfel1 2310 | . . . . 5 |
5 | fvmptf.4 | . . . . . . . 8 | |
6 | nfmpt1 4057 | . . . . . . . 8 | |
7 | 5, 6 | nfcxfr 2296 | . . . . . . 7 |
8 | 7, 2 | nffv 5478 | . . . . . 6 |
9 | 8, 3 | nfeq 2307 | . . . . 5 |
10 | 4, 9 | nfim 1552 | . . . 4 |
11 | fvmptf.3 | . . . . . 6 | |
12 | 11 | eleq1d 2226 | . . . . 5 |
13 | fveq2 5468 | . . . . . 6 | |
14 | 13, 11 | eqeq12d 2172 | . . . . 5 |
15 | 12, 14 | imbi12d 233 | . . . 4 |
16 | 5 | fvmpt2 5551 | . . . . 5 |
17 | 16 | ex 114 | . . . 4 |
18 | 2, 10, 15, 17 | vtoclgaf 2777 | . . 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 1335 wcel 2128 wnfc 2286 cvv 2712 cmpt 4025 cfv 5170 |
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 4082 ax-pow 4135 ax-pr 4169 |
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-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 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 |
This theorem is referenced by: fvmptd3 5561 elfvmptrab1 5562 sumrbdclem 11274 fsum3 11284 isumss 11288 prodrbdclem 11468 prodmodclem2a 11473 zproddc 11476 fprodntrivap 11481 prodssdc 11486 |
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