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Mirrors > Home > ILE Home > Th. List > dff13f | Unicode version |
Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.) |
Ref | Expression |
---|---|
dff13f.1 | |
dff13f.2 |
Ref | Expression |
---|---|
dff13f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff13 5407 | . 2 | |
2 | dff13f.2 | . . . . . . . . 9 | |
3 | nfcv 2178 | . . . . . . . . 9 | |
4 | 2, 3 | nffv 5185 | . . . . . . . 8 |
5 | nfcv 2178 | . . . . . . . . 9 | |
6 | 2, 5 | nffv 5185 | . . . . . . . 8 |
7 | 4, 6 | nfeq 2185 | . . . . . . 7 |
8 | nfv 1421 | . . . . . . 7 | |
9 | 7, 8 | nfim 1464 | . . . . . 6 |
10 | nfv 1421 | . . . . . 6 | |
11 | fveq2 5178 | . . . . . . . 8 | |
12 | 11 | eqeq2d 2051 | . . . . . . 7 |
13 | equequ2 1599 | . . . . . . 7 | |
14 | 12, 13 | imbi12d 223 | . . . . . 6 |
15 | 9, 10, 14 | cbvral 2529 | . . . . 5 |
16 | 15 | ralbii 2330 | . . . 4 |
17 | nfcv 2178 | . . . . . 6 | |
18 | dff13f.1 | . . . . . . . . 9 | |
19 | nfcv 2178 | . . . . . . . . 9 | |
20 | 18, 19 | nffv 5185 | . . . . . . . 8 |
21 | nfcv 2178 | . . . . . . . . 9 | |
22 | 18, 21 | nffv 5185 | . . . . . . . 8 |
23 | 20, 22 | nfeq 2185 | . . . . . . 7 |
24 | nfv 1421 | . . . . . . 7 | |
25 | 23, 24 | nfim 1464 | . . . . . 6 |
26 | 17, 25 | nfralxy 2360 | . . . . 5 |
27 | nfv 1421 | . . . . 5 | |
28 | fveq2 5178 | . . . . . . . 8 | |
29 | 28 | eqeq1d 2048 | . . . . . . 7 |
30 | equequ1 1598 | . . . . . . 7 | |
31 | 29, 30 | imbi12d 223 | . . . . . 6 |
32 | 31 | ralbidv 2326 | . . . . 5 |
33 | 26, 27, 32 | cbvral 2529 | . . . 4 |
34 | 16, 33 | bitri 173 | . . 3 |
35 | 34 | anbi2i 430 | . 2 |
36 | 1, 35 | bitri 173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wnfc 2165 wral 2306 wf 4898 wf1 4899 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fv 4910 |
This theorem is referenced by: f1mpt 5410 dom2lem 6252 |
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