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Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5518. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . . 3 | |
2 | 1 | ralimi 2495 | . 2 |
3 | fneq1 5211 | . . . . . . 7 | |
4 | eqid 2139 | . . . . . . . 8 | |
5 | 4 | mptfng 5248 | . . . . . . 7 |
6 | eqid 2139 | . . . . . . . 8 | |
7 | 6 | mptfng 5248 | . . . . . . 7 |
8 | 3, 5, 7 | 3bitr4g 222 | . . . . . 6 |
9 | 8 | biimpd 143 | . . . . 5 |
10 | r19.26 2558 | . . . . . . 7 | |
11 | nfmpt1 4021 | . . . . . . . . . 10 | |
12 | nfmpt1 4021 | . . . . . . . . . 10 | |
13 | 11, 12 | nfeq 2289 | . . . . . . . . 9 |
14 | simpll 518 | . . . . . . . . . . . 12 | |
15 | 14 | fveq1d 5423 | . . . . . . . . . . 11 |
16 | 4 | fvmpt2 5504 | . . . . . . . . . . . 12 |
17 | 16 | ad2ant2lr 501 | . . . . . . . . . . 11 |
18 | 6 | fvmpt2 5504 | . . . . . . . . . . . 12 |
19 | 18 | ad2ant2l 499 | . . . . . . . . . . 11 |
20 | 15, 17, 19 | 3eqtr3d 2180 | . . . . . . . . . 10 |
21 | 20 | exp31 361 | . . . . . . . . 9 |
22 | 13, 21 | ralrimi 2503 | . . . . . . . 8 |
23 | ralim 2491 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 10, 24 | syl5bir 152 | . . . . . 6 |
26 | 25 | expd 256 | . . . . 5 |
27 | 9, 26 | mpdd 41 | . . . 4 |
28 | 27 | com12 30 | . . 3 |
29 | eqid 2139 | . . . 4 | |
30 | mpteq12 4011 | . . . 4 | |
31 | 29, 30 | mpan 420 | . . 3 |
32 | 28, 31 | impbid1 141 | . 2 |
33 | 2, 32 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 cvv 2686 cmpt 3989 wfn 5118 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: eqfnfv 5518 eufnfv 5648 offveqb 6001 |
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