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Mirrors > Home > MPE Home > Th. List > brelrng | Structured version Visualization version GIF version |
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.) |
Ref | Expression |
---|---|
brelrng | ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5886 | . . . . 5 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵)) | |
2 | 1 | ancoms 457 | . . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵)) |
3 | 2 | biimp3ar 1466 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵◡𝐶𝐴) |
4 | breldmg 5916 | . . . 4 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶) | |
5 | 4 | 3com12 1120 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶) |
6 | 3, 5 | syld3an3 1406 | . 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ dom ◡𝐶) |
7 | df-rn 5693 | . 2 ⊢ ran 𝐶 = dom ◡𝐶 | |
8 | 6, 7 | eleqtrrdi 2840 | 1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5152 ◡ccnv 5681 dom cdm 5682 ran crn 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-cnv 5690 df-dm 5692 df-rn 5693 |
This theorem is referenced by: brelrn 5948 relelrn 5951 sossfld 6195 fvrn0 6932 pgpfaclem1 20045 perpln2 28535 |
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