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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrn3 | Structured version Visualization version GIF version |
Description: Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
Ref | Expression |
---|---|
elrn3 | ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5699 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵) |
3 | eldm3 35740 | . 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅) | |
4 | cnvxp 6178 | . . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V) | |
5 | 4 | ineq2i 4224 | . . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V)) |
6 | cnvin 6166 | . . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴})) | |
7 | df-res 5700 | . . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V)) | |
8 | 5, 6, 7 | 3eqtr4ri 2773 | . . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴})) |
9 | 8 | eqeq1i 2739 | . . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
10 | relinxp 5826 | . . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴})) | |
11 | cnveq0 6218 | . . . . 5 ⊢ (Rel (𝐵 ∩ (V × {𝐴})) → ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅)) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
13 | 9, 12 | bitr4i 278 | . . 3 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅) |
14 | 13 | necon3bii 2990 | . 2 ⊢ ((◡𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
15 | 2, 3, 14 | 3bitri 297 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∩ cin 3961 ∅c0 4338 {csn 4630 × cxp 5686 ◡ccnv 5687 dom cdm 5688 ran crn 5689 ↾ cres 5690 Rel wrel 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 |
This theorem is referenced by: (None) |
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