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Mirrors > Home > MPE Home > Th. List > relelrnb | Structured version Visualization version GIF version |
Description: Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.) |
Ref | Expression |
---|---|
relelrnb | ⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrng 5762 | . . 3 ⊢ (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) | |
2 | 1 | ibi 269 | . 2 ⊢ (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴) |
3 | relelrn 5815 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅) | |
4 | 3 | ex 415 | . . 3 ⊢ (Rel 𝑅 → (𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅)) |
5 | 4 | exlimdv 1934 | . 2 ⊢ (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅)) |
6 | 2, 5 | impbid2 228 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1780 ∈ wcel 2114 class class class wbr 5066 ran crn 5556 Rel wrel 5560 |
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-ral 3143 df-rex 3144 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-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: iscard4 39920 |
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