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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 | |
offval2.2 | |
offval2.3 | |
offval2.4 | |
offval2.5 |
Ref | Expression |
---|---|
ofrfval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval2.2 | . . . . . 6 | |
2 | 1 | ralrimiva 2543 | . . . . 5 |
3 | eqid 2170 | . . . . . 6 | |
4 | 3 | fnmpt 5322 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | offval2.4 | . . . . 5 | |
7 | 6 | fneq1d 5286 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | offval2.3 | . . . . . 6 | |
10 | 9 | ralrimiva 2543 | . . . . 5 |
11 | eqid 2170 | . . . . . 6 | |
12 | 11 | fnmpt 5322 | . . . . 5 |
13 | 10, 12 | syl 14 | . . . 4 |
14 | offval2.5 | . . . . 5 | |
15 | 14 | fneq1d 5286 | . . . 4 |
16 | 13, 15 | mpbird 166 | . . 3 |
17 | offval2.1 | . . 3 | |
18 | inidm 3336 | . . 3 | |
19 | 6 | adantr 274 | . . . 4 |
20 | 19 | fveq1d 5496 | . . 3 |
21 | 14 | adantr 274 | . . . 4 |
22 | 21 | fveq1d 5496 | . . 3 |
23 | 8, 16, 17, 17, 18, 20, 22 | ofrfval 6066 | . 2 |
24 | nffvmpt1 5505 | . . . . 5 | |
25 | nfcv 2312 | . . . . 5 | |
26 | nffvmpt1 5505 | . . . . 5 | |
27 | 24, 25, 26 | nfbr 4033 | . . . 4 |
28 | nfv 1521 | . . . 4 | |
29 | fveq2 5494 | . . . . 5 | |
30 | fveq2 5494 | . . . . 5 | |
31 | 29, 30 | breq12d 4000 | . . . 4 |
32 | 27, 28, 31 | cbvral 2692 | . . 3 |
33 | simpr 109 | . . . . . 6 | |
34 | 3 | fvmpt2 5577 | . . . . . 6 |
35 | 33, 1, 34 | syl2anc 409 | . . . . 5 |
36 | 11 | fvmpt2 5577 | . . . . . 6 |
37 | 33, 9, 36 | syl2anc 409 | . . . . 5 |
38 | 35, 37 | breq12d 4000 | . . . 4 |
39 | 38 | ralbidva 2466 | . . 3 |
40 | 32, 39 | syl5bb 191 | . 2 |
41 | 23, 40 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 class class class wbr 3987 cmpt 4048 wfn 5191 cfv 5196 cofr 6057 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ofr 6059 |
This theorem is referenced by: (None) |
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