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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 5276 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) | |
4 | 1, 2, 3 | mp3an12 1406 | 1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ran crn 5039 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 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-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: opelrn 5278 dfco2a 5552 cores 5555 dffun9 5832 funcnv 5872 rntpos 7252 aceq3lem 8826 axdclem 9224 axdclem2 9225 cotr2g 13563 shftfval 13658 psdmrn 17030 metustexhalf 22171 itg1addlem4 23272 |
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