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Mirrors > Home > MPE Home > Th. List > cbvmptf | Structured version Visualization version 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 Thierry Arnoux, 9-Mar-2017.) |
Ref | Expression |
---|---|
cbvmptf.1 | |
cbvmptf.2 | |
cbvmptf.3 | |
cbvmptf.4 | |
cbvmptf.5 |
Ref | Expression |
---|---|
cbvmptf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . . . 4 | |
2 | cbvmptf.1 | . . . . . 6 | |
3 | 2 | nfcri 2758 | . . . . 5 |
4 | nfs1v 2437 | . . . . 5 | |
5 | 3, 4 | nfan 1828 | . . . 4 |
6 | eleq1 2689 | . . . . 5 | |
7 | sbequ12 2111 | . . . . 5 | |
8 | 6, 7 | anbi12d 747 | . . . 4 |
9 | 1, 5, 8 | cbvopab1 4723 | . . 3 |
10 | cbvmptf.2 | . . . . . 6 | |
11 | 10 | nfcri 2758 | . . . . 5 |
12 | cbvmptf.3 | . . . . . . 7 | |
13 | 12 | nfeq2 2780 | . . . . . 6 |
14 | 13 | nfsb 2440 | . . . . 5 |
15 | 11, 14 | nfan 1828 | . . . 4 |
16 | nfv 1843 | . . . 4 | |
17 | eleq1 2689 | . . . . 5 | |
18 | cbvmptf.4 | . . . . . . 7 | |
19 | 18 | nfeq2 2780 | . . . . . 6 |
20 | cbvmptf.5 | . . . . . . 7 | |
21 | 20 | eqeq2d 2632 | . . . . . 6 |
22 | 19, 21 | sbhypf 3253 | . . . . 5 |
23 | 17, 22 | anbi12d 747 | . . . 4 |
24 | 15, 16, 23 | cbvopab1 4723 | . . 3 |
25 | 9, 24 | eqtri 2644 | . 2 |
26 | df-mpt 4730 | . 2 | |
27 | df-mpt 4730 | . 2 | |
28 | 25, 26, 27 | 3eqtr4i 2654 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wsb 1880 wcel 1990 wnfc 2751 copab 4712 cmpt 4729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-mpt 4730 |
This theorem is referenced by: resmptf 5451 fvmpt2f 6283 offval2f 6909 numclwlk1lem2 27230 suppss2f 29439 fmptdF 29456 acunirnmpt2f 29461 funcnv4mpt 29470 cbvesum 30104 esumpfinvalf 30138 binomcxplemdvbinom 38552 binomcxplemdvsum 38554 binomcxplemnotnn0 38555 supxrleubrnmptf 39680 fnlimfv 39895 fnlimfvre2 39909 fnlimf 39910 limsupequzmptf 39963 sge0iunmptlemre 40632 smflim 40985 smflim2 41012 smfsup 41020 smfinf 41024 smflimsuplem2 41027 smflimsuplem5 41030 smflimsup 41034 smfliminf 41037 |
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