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Mirrors > Home > MPE Home > Th. List > eqrelrdv | Structured version Visualization version Unicode version |
Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.) |
Ref | Expression |
---|---|
eqrelrdv.1 | |
eqrelrdv.2 | |
eqrelrdv.3 |
Ref | Expression |
---|---|
eqrelrdv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelrdv.3 | . . 3 | |
2 | 1 | alrimivv 1856 | . 2 |
3 | eqrelrdv.1 | . . 3 | |
4 | eqrelrdv.2 | . . 3 | |
5 | eqrel 5209 | . . 3 | |
6 | 3, 4, 5 | mp2an 708 | . 2 |
7 | 2, 6 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wal 1481 wceq 1483 wcel 1990 cop 4183 wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 df-opab 4713 df-xp 5120 df-rel 5121 |
This theorem is referenced by: eqbrrdiv 5218 fcnvres 6082 fmptco 6396 fpwwe2lem8 9459 fpwwe2lem12 9463 fsumcom2 14505 fsumcom2OLD 14506 fprodcom2 14714 fprodcom2OLD 14715 gsumcom2 18374 lgsquadlem1 25105 lgsquadlem2 25106 fmptcof2 29457 dfcnv2 29476 dih1dimatlem 36618 |
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