Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > seex | Structured version Visualization version Unicode version |
Description: The -preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.) |
Ref | Expression |
---|---|
seex | Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-se 5074 | . 2 Se | |
2 | breq2 4657 | . . . . 5 | |
3 | 2 | rabbidv 3189 | . . . 4 |
4 | 3 | eleq1d 2686 | . . 3 |
5 | 4 | rspccva 3308 | . 2 |
6 | 1, 5 | sylanb 489 | 1 Se |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 crab 2916 cvv 3200 class class class wbr 4653 Se wse 5071 |
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-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-se 5074 |
This theorem is referenced by: wereu2 5111 setlikespec 5701 fnse 7294 ordtypelem10 8432 |
Copyright terms: Public domain | W3C validator |