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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 5236 | . . 3 ⊢ (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) | |
2 | 1 | ibi 255 | . 2 ⊢ (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴) |
3 | relelrn 5280 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅) | |
4 | 3 | ex 449 | . . 3 ⊢ (Rel 𝑅 → (𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅)) |
5 | 4 | exlimdv 1848 | . 2 ⊢ (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅)) |
6 | 2, 5 | impbid2 215 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∃wex 1695 ∈ wcel 1977 class class class wbr 4583 ran crn 5039 Rel wrel 5043 |
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-ral 2901 df-rex 2902 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-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: (None) |
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