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Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5511. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . . 3 | |
2 | 1 | ralimi 2493 | . 2 |
3 | fneq1 5206 | . . . . . . 7 | |
4 | eqid 2137 | . . . . . . . 8 | |
5 | 4 | mptfng 5243 | . . . . . . 7 |
6 | eqid 2137 | . . . . . . . 8 | |
7 | 6 | mptfng 5243 | . . . . . . 7 |
8 | 3, 5, 7 | 3bitr4g 222 | . . . . . 6 |
9 | 8 | biimpd 143 | . . . . 5 |
10 | r19.26 2556 | . . . . . . 7 | |
11 | nfmpt1 4016 | . . . . . . . . . 10 | |
12 | nfmpt1 4016 | . . . . . . . . . 10 | |
13 | 11, 12 | nfeq 2287 | . . . . . . . . 9 |
14 | simpll 518 | . . . . . . . . . . . 12 | |
15 | 14 | fveq1d 5416 | . . . . . . . . . . 11 |
16 | 4 | fvmpt2 5497 | . . . . . . . . . . . 12 |
17 | 16 | ad2ant2lr 501 | . . . . . . . . . . 11 |
18 | 6 | fvmpt2 5497 | . . . . . . . . . . . 12 |
19 | 18 | ad2ant2l 499 | . . . . . . . . . . 11 |
20 | 15, 17, 19 | 3eqtr3d 2178 | . . . . . . . . . 10 |
21 | 20 | exp31 361 | . . . . . . . . 9 |
22 | 13, 21 | ralrimi 2501 | . . . . . . . 8 |
23 | ralim 2489 | . . . . . . . 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 2137 | . . . 4 | |
30 | mpteq12 4006 | . . . 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 2414 cvv 2681 cmpt 3984 wfn 5113 cfv 5118 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: eqfnfv 5511 eufnfv 5641 offveqb 5994 |
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