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Mirrors > Home > MPE Home > Th. List > rnsnopg | Structured version Visualization version Unicode version |
Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
rnsnopg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5125 | . . 3 | |
2 | dfdm4 5316 | . . . 4 | |
3 | df-rn 5125 | . . . 4 | |
4 | cnvcnvsn 5612 | . . . . 5 | |
5 | 4 | dmeqi 5325 | . . . 4 |
6 | 2, 3, 5 | 3eqtri 2648 | . . 3 |
7 | 1, 6 | eqtr4i 2647 | . 2 |
8 | dmsnopg 5606 | . 2 | |
9 | 7, 8 | syl5eq 2668 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 csn 4177 cop 4183 ccnv 5113 cdm 5114 crn 5115 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 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-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: rnpropg 5615 rnsnop 5616 funcnvpr 5950 funcnvtp 5951 dprdsn 18435 usgr1e 26137 1loopgredg 26397 1egrvtxdg0 26407 uspgrloopedg 26414 noextend 31819 rnsnf 39370 |
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