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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 5759 | . 2 | |
2 | dff13f.2 | . . . . . . . . 9 | |
3 | nfcv 2317 | . . . . . . . . 9 | |
4 | 2, 3 | nffv 5517 | . . . . . . . 8 |
5 | nfcv 2317 | . . . . . . . . 9 | |
6 | 2, 5 | nffv 5517 | . . . . . . . 8 |
7 | 4, 6 | nfeq 2325 | . . . . . . 7 |
8 | nfv 1526 | . . . . . . 7 | |
9 | 7, 8 | nfim 1570 | . . . . . 6 |
10 | nfv 1526 | . . . . . 6 | |
11 | fveq2 5507 | . . . . . . . 8 | |
12 | 11 | eqeq2d 2187 | . . . . . . 7 |
13 | equequ2 1711 | . . . . . . 7 | |
14 | 12, 13 | imbi12d 234 | . . . . . 6 |
15 | 9, 10, 14 | cbvral 2697 | . . . . 5 |
16 | 15 | ralbii 2481 | . . . 4 |
17 | nfcv 2317 | . . . . . 6 | |
18 | dff13f.1 | . . . . . . . . 9 | |
19 | nfcv 2317 | . . . . . . . . 9 | |
20 | 18, 19 | nffv 5517 | . . . . . . . 8 |
21 | nfcv 2317 | . . . . . . . . 9 | |
22 | 18, 21 | nffv 5517 | . . . . . . . 8 |
23 | 20, 22 | nfeq 2325 | . . . . . . 7 |
24 | nfv 1526 | . . . . . . 7 | |
25 | 23, 24 | nfim 1570 | . . . . . 6 |
26 | 17, 25 | nfralxy 2513 | . . . . 5 |
27 | nfv 1526 | . . . . 5 | |
28 | fveq2 5507 | . . . . . . . 8 | |
29 | 28 | eqeq1d 2184 | . . . . . . 7 |
30 | equequ1 1710 | . . . . . . 7 | |
31 | 29, 30 | imbi12d 234 | . . . . . 6 |
32 | 31 | ralbidv 2475 | . . . . 5 |
33 | 26, 27, 32 | cbvral 2697 | . . . 4 |
34 | 16, 33 | bitri 184 | . . 3 |
35 | 34 | anbi2i 457 | . 2 |
36 | 1, 35 | bitri 184 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wnfc 2304 wral 2453 wf 5204 wf1 5205 cfv 5208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fv 5216 |
This theorem is referenced by: f1mpt 5762 dom2lem 6762 |
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