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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 5696 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵) |
| 3 | eldm3 35761 | . 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅) | |
| 4 | cnvxp 6177 | . . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V) | |
| 5 | 4 | ineq2i 4217 | . . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V)) |
| 6 | cnvin 6164 | . . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴})) | |
| 7 | df-res 5697 | . . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V)) | |
| 8 | 5, 6, 7 | 3eqtr4ri 2776 | . . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴})) |
| 9 | 8 | eqeq1i 2742 | . . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
| 10 | relinxp 5824 | . . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴})) | |
| 11 | cnveq0 6217 | . . . . 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 2993 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∩ cin 3950 ∅c0 4333 {csn 4626 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 Rel wrel 5690 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 |
| This theorem is referenced by: (None) |
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