Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5265. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2566 | . . 3 | |
2 | 1 | ralimi 2384 | . 2 |
3 | fneq1 4987 | . . . . . . 7 | |
4 | eqid 2040 | . . . . . . . 8 | |
5 | 4 | mptfng 5024 | . . . . . . 7 |
6 | eqid 2040 | . . . . . . . 8 | |
7 | 6 | mptfng 5024 | . . . . . . 7 |
8 | 3, 5, 7 | 3bitr4g 212 | . . . . . 6 |
9 | 8 | biimpd 132 | . . . . 5 |
10 | r19.26 2441 | . . . . . . 7 | |
11 | nfmpt1 3850 | . . . . . . . . . 10 | |
12 | nfmpt1 3850 | . . . . . . . . . 10 | |
13 | 11, 12 | nfeq 2185 | . . . . . . . . 9 |
14 | simpll 481 | . . . . . . . . . . . 12 | |
15 | 14 | fveq1d 5180 | . . . . . . . . . . 11 |
16 | 4 | fvmpt2 5254 | . . . . . . . . . . . 12 |
17 | 16 | ad2ant2lr 479 | . . . . . . . . . . 11 |
18 | 6 | fvmpt2 5254 | . . . . . . . . . . . 12 |
19 | 18 | ad2ant2l 477 | . . . . . . . . . . 11 |
20 | 15, 17, 19 | 3eqtr3d 2080 | . . . . . . . . . 10 |
21 | 20 | exp31 346 | . . . . . . . . 9 |
22 | 13, 21 | ralrimi 2390 | . . . . . . . 8 |
23 | ralim 2380 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 10, 24 | syl5bir 142 | . . . . . 6 |
26 | 25 | expd 245 | . . . . 5 |
27 | 9, 26 | mpdd 36 | . . . 4 |
28 | 27 | com12 27 | . . 3 |
29 | eqid 2040 | . . . 4 | |
30 | mpteq12 3840 | . . . 4 | |
31 | 29, 30 | mpan 400 | . . 3 |
32 | 28, 31 | impbid1 130 | . 2 |
33 | 2, 32 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 cmpt 3818 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: eqfnfv 5265 eufnfv 5389 offveqb 5730 |
Copyright terms: Public domain | W3C validator |