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Mirrors > Home > MPE Home > Th. List > brrelex2i | Structured version Visualization version GIF version |
Description: The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brrelexi.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
brrelex2i | ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelexi.1 | . 2 ⊢ Rel 𝑅 | |
2 | brrelex2 5081 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | |
3 | 1, 2 | mpan 702 | 1 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: vtoclr 5086 brdomi 7852 domdifsn 7928 undom 7933 xpdom2 7940 xpdom1g 7942 domunsncan 7945 enfixsn 7954 fodomr 7996 pwdom 7997 domssex 8006 xpen 8008 mapdom1 8010 mapdom2 8016 pwen 8018 sucdom2 8041 unxpdom 8052 unxpdom2 8053 sucxpdom 8054 isfinite2 8103 infn0 8107 fin2inf 8108 fsuppimp 8164 suppeqfsuppbi 8172 fsuppsssupp 8174 fsuppunbi 8179 funsnfsupp 8182 mapfien2 8197 wemapso2 8341 card2on 8342 elharval 8351 harword 8353 brwdomi 8356 brwdomn0 8357 domwdom 8362 wdomtr 8363 wdompwdom 8366 canthwdom 8367 brwdom3i 8371 unwdomg 8372 xpwdomg 8373 unxpwdom 8377 infdifsn 8437 infdiffi 8438 isnum2 8654 wdomfil 8767 cdaen 8878 cdaenun 8879 cdadom1 8891 cdaxpdom 8894 cdainf 8897 infcda1 8898 pwcdaidm 8900 cdalepw 8901 infpss 8922 infmap2 8923 fictb 8950 infpssALT 9018 enfin2i 9026 fin34 9095 fodomb 9229 wdomac 9230 iundom2g 9241 iundom 9243 sdomsdomcard 9261 infxpidm 9263 engch 9329 fpwwe2lem3 9334 canthp1lem1 9353 canthp1lem2 9354 canthp1 9355 pwfseq 9365 pwxpndom2 9366 pwxpndom 9367 pwcdandom 9368 hargch 9374 gchaclem 9379 hasheni 12998 hashdomi 13030 brfi1indALT 13137 brfi1indALTOLD 13143 clim 14073 rlim 14074 ntrivcvgn0 14469 ssc1 16304 ssc2 16305 ssctr 16308 frgpnabl 18101 dprddomprc 18222 dprdval 18225 dprdgrp 18227 dprdf 18228 dprdssv 18238 subgdmdprd 18256 dprd2da 18264 1stcrestlem 21065 hauspwdom 21114 isref 21122 ufilen 21544 dvle 23574 uhgrav 25825 ushgraf 25831 ushgrauhgra 25832 umgraf2 25846 umgrares 25853 umisuhgra 25856 umgraun 25857 uslgrav 25866 usgrav 25867 uslgraf 25874 iscusgra0 25986 frisusgrapr 26518 locfinref 29236 isfne4 31505 fnetr 31516 topfneec 31520 fnessref 31522 refssfne 31523 phpreu 32563 climf 38689 climf2 38733 subgrv 40494 linindsv 42028 |
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