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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 5753 | . . . . 5 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵)) | |
2 | 1 | ancoms 461 | . . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵)) |
3 | 2 | biimp3ar 1466 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵◡𝐶𝐴) |
4 | breldmg 5781 | . . . 4 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶) | |
5 | 4 | 3com12 1119 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶) |
6 | 3, 5 | syld3an3 1405 | . 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ dom ◡𝐶) |
7 | df-rn 5569 | . 2 ⊢ ran 𝐶 = dom ◡𝐶 | |
8 | 6, 7 | eleqtrrdi 2927 | 1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2113 class class class wbr 5069 ◡ccnv 5557 dom cdm 5558 ran crn 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-cnv 5566 df-dm 5568 df-rn 5569 |
This theorem is referenced by: brelrn 5815 relelrn 5818 sossfld 6046 fvrn0 6701 pgpfaclem1 19206 perpln2 26500 |
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