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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 5269 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 2610 | . . 3 | |
2 | fvmptf.1 | . . . 4 | |
3 | fvmptf.2 | . . . . . 6 | |
4 | 3 | nfel1 2229 | . . . . 5 |
5 | fvmptf.4 | . . . . . . . 8 | |
6 | nfmpt1 3871 | . . . . . . . 8 | |
7 | 5, 6 | nfcxfr 2216 | . . . . . . 7 |
8 | 7, 2 | nffv 5205 | . . . . . 6 |
9 | 8, 3 | nfeq 2226 | . . . . 5 |
10 | 4, 9 | nfim 1504 | . . . 4 |
11 | fvmptf.3 | . . . . . 6 | |
12 | 11 | eleq1d 2147 | . . . . 5 |
13 | fveq2 5198 | . . . . . 6 | |
14 | 13, 11 | eqeq12d 2095 | . . . . 5 |
15 | 12, 14 | imbi12d 232 | . . . 4 |
16 | 5 | fvmpt2 5275 | . . . . 5 |
17 | 16 | ex 113 | . . . 4 |
18 | 2, 10, 15, 17 | vtoclgaf 2663 | . . 3 |
19 | 1, 18 | syl5 32 | . 2 |
20 | 19 | imp 122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wnfc 2206 cvv 2601 cmpt 3839 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 |
This theorem is referenced by: (None) |
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