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Mirrors > Home > ILE Home > Th. List > cnvopab | Unicode version |
Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvopab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4989 | . 2 | |
2 | relopab 4738 | . 2 | |
3 | opelopabsbALT 4244 | . . . 4 | |
4 | sbcom2 1980 | . . . 4 | |
5 | 3, 4 | bitri 183 | . . 3 |
6 | vex 2733 | . . . 4 | |
7 | vex 2733 | . . . 4 | |
8 | 6, 7 | opelcnv 4793 | . . 3 |
9 | opelopabsbALT 4244 | . . 3 | |
10 | 5, 8, 9 | 3bitr4i 211 | . 2 |
11 | 1, 2, 10 | eqrelriiv 4705 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wsb 1755 wcel 2141 cop 3586 copab 4049 ccnv 4610 |
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-sep 4107 ax-pow 4160 ax-pr 4194 |
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-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 |
This theorem is referenced by: mptcnv 5013 cnvxp 5029 mptpreima 5104 f1ocnvd 6051 cnvoprab 6213 mapsncnv 6673 |
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