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Mirrors > Home > ILE Home > Th. List > ofrfval2 | Unicode version |
Description: The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval2.1 |
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offval2.2 |
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offval2.3 |
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offval2.4 |
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offval2.5 |
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Ref | Expression |
---|---|
ofrfval2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval2.2 |
. . . . . 6
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2 | 1 | ralrimiva 2447 |
. . . . 5
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3 | eqid 2089 |
. . . . . 6
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4 | 3 | fnmpt 5153 |
. . . . 5
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5 | 2, 4 | syl 14 |
. . . 4
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6 | offval2.4 |
. . . . 5
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7 | 6 | fneq1d 5117 |
. . . 4
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8 | 5, 7 | mpbird 166 |
. . 3
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9 | offval2.3 |
. . . . . 6
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10 | 9 | ralrimiva 2447 |
. . . . 5
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11 | eqid 2089 |
. . . . . 6
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12 | 11 | fnmpt 5153 |
. . . . 5
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13 | 10, 12 | syl 14 |
. . . 4
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14 | offval2.5 |
. . . . 5
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15 | 14 | fneq1d 5117 |
. . . 4
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16 | 13, 15 | mpbird 166 |
. . 3
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17 | offval2.1 |
. . 3
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18 | inidm 3210 |
. . 3
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19 | 6 | adantr 271 |
. . . 4
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20 | 19 | fveq1d 5320 |
. . 3
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21 | 14 | adantr 271 |
. . . 4
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22 | 21 | fveq1d 5320 |
. . 3
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23 | 8, 16, 17, 17, 18, 20, 22 | ofrfval 5878 |
. 2
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24 | nffvmpt1 5329 |
. . . . 5
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25 | nfcv 2229 |
. . . . 5
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26 | nffvmpt1 5329 |
. . . . 5
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27 | 24, 25, 26 | nfbr 3895 |
. . . 4
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28 | nfv 1467 |
. . . 4
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29 | fveq2 5318 |
. . . . 5
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30 | fveq2 5318 |
. . . . 5
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31 | 29, 30 | breq12d 3864 |
. . . 4
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32 | 27, 28, 31 | cbvral 2587 |
. . 3
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33 | simpr 109 |
. . . . . 6
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34 | 3 | fvmpt2 5399 |
. . . . . 6
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35 | 33, 1, 34 | syl2anc 404 |
. . . . 5
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36 | 11 | fvmpt2 5399 |
. . . . . 6
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37 | 33, 9, 36 | syl2anc 404 |
. . . . 5
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38 | 35, 37 | breq12d 3864 |
. . . 4
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39 | 38 | ralbidva 2377 |
. . 3
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40 | 32, 39 | syl5bb 191 |
. 2
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41 | 23, 40 | bitrd 187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ofr 5871 |
This theorem is referenced by: (None) |
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