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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 5669 | . 2 | |
2 | dff13f.2 | . . . . . . . . 9 | |
3 | nfcv 2281 | . . . . . . . . 9 | |
4 | 2, 3 | nffv 5431 | . . . . . . . 8 |
5 | nfcv 2281 | . . . . . . . . 9 | |
6 | 2, 5 | nffv 5431 | . . . . . . . 8 |
7 | 4, 6 | nfeq 2289 | . . . . . . 7 |
8 | nfv 1508 | . . . . . . 7 | |
9 | 7, 8 | nfim 1551 | . . . . . 6 |
10 | nfv 1508 | . . . . . 6 | |
11 | fveq2 5421 | . . . . . . . 8 | |
12 | 11 | eqeq2d 2151 | . . . . . . 7 |
13 | equequ2 1689 | . . . . . . 7 | |
14 | 12, 13 | imbi12d 233 | . . . . . 6 |
15 | 9, 10, 14 | cbvral 2650 | . . . . 5 |
16 | 15 | ralbii 2441 | . . . 4 |
17 | nfcv 2281 | . . . . . 6 | |
18 | dff13f.1 | . . . . . . . . 9 | |
19 | nfcv 2281 | . . . . . . . . 9 | |
20 | 18, 19 | nffv 5431 | . . . . . . . 8 |
21 | nfcv 2281 | . . . . . . . . 9 | |
22 | 18, 21 | nffv 5431 | . . . . . . . 8 |
23 | 20, 22 | nfeq 2289 | . . . . . . 7 |
24 | nfv 1508 | . . . . . . 7 | |
25 | 23, 24 | nfim 1551 | . . . . . 6 |
26 | 17, 25 | nfralxy 2471 | . . . . 5 |
27 | nfv 1508 | . . . . 5 | |
28 | fveq2 5421 | . . . . . . . 8 | |
29 | 28 | eqeq1d 2148 | . . . . . . 7 |
30 | equequ1 1688 | . . . . . . 7 | |
31 | 29, 30 | imbi12d 233 | . . . . . 6 |
32 | 31 | ralbidv 2437 | . . . . 5 |
33 | 26, 27, 32 | cbvral 2650 | . . . 4 |
34 | 16, 33 | bitri 183 | . . 3 |
35 | 34 | anbi2i 452 | . 2 |
36 | 1, 35 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wnfc 2268 wral 2416 wf 5119 wf1 5120 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fv 5131 |
This theorem is referenced by: f1mpt 5672 dom2lem 6666 |
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