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Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5286. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2610 | . . 3 | |
2 | 1 | ralimi 2426 | . 2 |
3 | fneq1 5007 | . . . . . . 7 | |
4 | eqid 2081 | . . . . . . . 8 | |
5 | 4 | mptfng 5044 | . . . . . . 7 |
6 | eqid 2081 | . . . . . . . 8 | |
7 | 6 | mptfng 5044 | . . . . . . 7 |
8 | 3, 5, 7 | 3bitr4g 221 | . . . . . 6 |
9 | 8 | biimpd 142 | . . . . 5 |
10 | r19.26 2485 | . . . . . . 7 | |
11 | nfmpt1 3871 | . . . . . . . . . 10 | |
12 | nfmpt1 3871 | . . . . . . . . . 10 | |
13 | 11, 12 | nfeq 2226 | . . . . . . . . 9 |
14 | simpll 495 | . . . . . . . . . . . 12 | |
15 | 14 | fveq1d 5200 | . . . . . . . . . . 11 |
16 | 4 | fvmpt2 5275 | . . . . . . . . . . . 12 |
17 | 16 | ad2ant2lr 493 | . . . . . . . . . . 11 |
18 | 6 | fvmpt2 5275 | . . . . . . . . . . . 12 |
19 | 18 | ad2ant2l 491 | . . . . . . . . . . 11 |
20 | 15, 17, 19 | 3eqtr3d 2121 | . . . . . . . . . 10 |
21 | 20 | exp31 356 | . . . . . . . . 9 |
22 | 13, 21 | ralrimi 2432 | . . . . . . . 8 |
23 | ralim 2422 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 10, 24 | syl5bir 151 | . . . . . 6 |
26 | 25 | expd 254 | . . . . 5 |
27 | 9, 26 | mpdd 40 | . . . 4 |
28 | 27 | com12 30 | . . 3 |
29 | eqid 2081 | . . . 4 | |
30 | mpteq12 3861 | . . . 4 | |
31 | 29, 30 | mpan 414 | . . 3 |
32 | 28, 31 | impbid1 140 | . 2 |
33 | 2, 32 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 cvv 2601 cmpt 3839 wfn 4917 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: eqfnfv 5286 eufnfv 5410 offveqb 5750 |
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