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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 5568 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵) |
3 | eldm3 32999 | . 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅) | |
4 | cnvxp 6016 | . . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V) | |
5 | 4 | ineq2i 4188 | . . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V)) |
6 | cnvin 6005 | . . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴})) | |
7 | df-res 5569 | . . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V)) | |
8 | 5, 6, 7 | 3eqtr4ri 2857 | . . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴})) |
9 | 8 | eqeq1i 2828 | . . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
10 | relinxp 5689 | . . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴})) | |
11 | cnveq0 6056 | . . . . 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 280 | . . 3 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅) |
14 | 13 | necon3bii 3070 | . 2 ⊢ ((◡𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
15 | 2, 3, 14 | 3bitri 299 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∩ cin 3937 ∅c0 4293 {csn 4569 × cxp 5555 ◡ccnv 5556 dom cdm 5557 ran crn 5558 ↾ cres 5559 Rel wrel 5562 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 |
This theorem is referenced by: (None) |
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