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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 2560 |
. . . . 5
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3 | eqid 2187 |
. . . . . 6
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4 | 3 | fnmpt 5354 |
. . . . 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 5318 |
. . . 4
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8 | 5, 7 | mpbird 167 |
. . 3
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9 | offval2.3 |
. . . . . 6
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10 | 9 | ralrimiva 2560 |
. . . . 5
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11 | eqid 2187 |
. . . . . 6
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12 | 11 | fnmpt 5354 |
. . . . 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 5318 |
. . . 4
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16 | 13, 15 | mpbird 167 |
. . 3
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17 | offval2.1 |
. . 3
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18 | inidm 3356 |
. . 3
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19 | 6 | adantr 276 |
. . . 4
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20 | 19 | fveq1d 5529 |
. . 3
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21 | 14 | adantr 276 |
. . . 4
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22 | 21 | fveq1d 5529 |
. . 3
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23 | 8, 16, 17, 17, 18, 20, 22 | ofrfval 6105 |
. 2
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24 | nffvmpt1 5538 |
. . . . 5
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25 | nfcv 2329 |
. . . . 5
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26 | nffvmpt1 5538 |
. . . . 5
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27 | 24, 25, 26 | nfbr 4061 |
. . . 4
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28 | nfv 1538 |
. . . 4
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29 | fveq2 5527 |
. . . . 5
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30 | fveq2 5527 |
. . . . 5
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31 | 29, 30 | breq12d 4028 |
. . . 4
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32 | 27, 28, 31 | cbvral 2711 |
. . 3
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33 | simpr 110 |
. . . . . 6
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34 | 3 | fvmpt2 5612 |
. . . . . 6
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35 | 33, 1, 34 | syl2anc 411 |
. . . . 5
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36 | 11 | fvmpt2 5612 |
. . . . . 6
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37 | 33, 9, 36 | syl2anc 411 |
. . . . 5
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38 | 35, 37 | breq12d 4028 |
. . . 4
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39 | 38 | ralbidva 2483 |
. . 3
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40 | 32, 39 | bitrid 192 |
. 2
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41 | 23, 40 | bitrd 188 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ofr 6098 |
This theorem is referenced by: (None) |
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