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Mirrors > Home > ILE Home > Th. List > cbvmpt | Unicode version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) |
Ref | Expression |
---|---|
cbvmpt.1 | |
cbvmpt.2 | |
cbvmpt.3 |
Ref | Expression |
---|---|
cbvmpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . . 4 | |
2 | nfv 1508 | . . . . 5 | |
3 | nfs1v 1912 | . . . . 5 | |
4 | 2, 3 | nfan 1544 | . . . 4 |
5 | eleq1 2202 | . . . . 5 | |
6 | sbequ12 1744 | . . . . 5 | |
7 | 5, 6 | anbi12d 464 | . . . 4 |
8 | 1, 4, 7 | cbvopab1 4001 | . . 3 |
9 | nfv 1508 | . . . . 5 | |
10 | cbvmpt.1 | . . . . . . 7 | |
11 | 10 | nfeq2 2293 | . . . . . 6 |
12 | 11 | nfsb 1919 | . . . . 5 |
13 | 9, 12 | nfan 1544 | . . . 4 |
14 | nfv 1508 | . . . 4 | |
15 | eleq1 2202 | . . . . 5 | |
16 | sbequ 1812 | . . . . . 6 | |
17 | cbvmpt.2 | . . . . . . . 8 | |
18 | 17 | nfeq2 2293 | . . . . . . 7 |
19 | cbvmpt.3 | . . . . . . . 8 | |
20 | 19 | eqeq2d 2151 | . . . . . . 7 |
21 | 18, 20 | sbie 1764 | . . . . . 6 |
22 | 16, 21 | syl6bb 195 | . . . . 5 |
23 | 15, 22 | anbi12d 464 | . . . 4 |
24 | 13, 14, 23 | cbvopab1 4001 | . . 3 |
25 | 8, 24 | eqtri 2160 | . 2 |
26 | df-mpt 3991 | . 2 | |
27 | df-mpt 3991 | . 2 | |
28 | 25, 26, 27 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wsb 1735 wnfc 2268 copab 3988 cmpt 3989 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-mpt 3991 |
This theorem is referenced by: cbvmptv 4024 dffn5imf 5476 fvmpts 5499 fvmpt2 5504 mptfvex 5506 fmptcof 5587 fmptcos 5588 fliftfuns 5699 offval2 5997 qliftfuns 6513 summodclem2a 11150 zsumdc 11153 fsum3cvg2 11163 cbvprod 11327 cnmpt1t 12454 fsumcncntop 12725 limcmpted 12801 |
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