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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 5811 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) | |
4 | 1, 2, 3 | mp3an12 1447 | 1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: opelrn 5813 dfco2a 6099 cores 6102 dffun9 6384 funcnv 6423 rntpos 7905 aceq3lem 9546 axdclem 9941 axdclem2 9942 cotr2g 14336 shftfval 14429 psdmrn 17817 metustexhalf 23166 itg1addlem4 24300 |
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