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Mirrors > Home > MPE Home > Th. List > brelrn | Structured version Visualization version GIF version |
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
brelrn.1 | ⊢ 𝐴 ∈ V |
brelrn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brelrn | ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brelrn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | brelrn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | brelrng 5775 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) | |
4 | 1, 2, 3 | mp3an12 1448 | 1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: opelrn 5777 dfco2a 6066 cores 6069 dffun9 6353 funcnv 6393 rntpos 7888 aceq3lem 9531 axdclem 9930 axdclem2 9931 cotr2g 14327 shftfval 14421 psdmrn 17809 metustexhalf 23163 itg1addlem4 24303 |
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