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Mathbox for Scott Fenton |
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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 5197 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
2 | 1 | eleq2i 2763 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵) |
3 | eldm3 31847 | . 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅) | |
4 | cnvxp 5629 | . . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V) | |
5 | 4 | ineq2i 3887 | . . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V)) |
6 | cnvin 5618 | . . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴})) | |
7 | df-res 5198 | . . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V)) | |
8 | 5, 6, 7 | 3eqtr4ri 2725 | . . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴})) |
9 | 8 | eqeq1i 2697 | . . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
10 | inss2 3910 | . . . . . 6 ⊢ (𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴}) | |
11 | relxp 5203 | . . . . . 6 ⊢ Rel (V × {𝐴}) | |
12 | relss 5283 | . . . . . 6 ⊢ ((𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴}) → (Rel (V × {𝐴}) → Rel (𝐵 ∩ (V × {𝐴})))) | |
13 | 10, 11, 12 | mp2 9 | . . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴})) |
14 | cnveq0 5669 | . . . . 5 ⊢ (Rel (𝐵 ∩ (V × {𝐴})) → ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
16 | 9, 15 | bitr4i 267 | . . 3 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅) |
17 | 16 | necon3bii 2916 | . 2 ⊢ ((◡𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
18 | 2, 3, 17 | 3bitri 286 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1564 ∈ wcel 2071 ≠ wne 2864 Vcvv 3272 ∩ cin 3647 ⊆ wss 3648 ∅c0 3991 {csn 4253 × cxp 5184 ◡ccnv 5185 dom cdm 5186 ran crn 5187 ↾ cres 5188 Rel wrel 5191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pr 4979 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-ral 2987 df-rex 2988 df-rab 2991 df-v 3274 df-sbc 3510 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-nul 3992 df-if 4163 df-sn 4254 df-pr 4256 df-op 4260 df-br 4729 df-opab 4789 df-xp 5192 df-rel 5193 df-cnv 5194 df-dm 5196 df-rn 5197 df-res 5198 |
This theorem is referenced by: (None) |
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