Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 5125 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵) |
3 | eldm3 31651 | . 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅) | |
4 | cnvxp 5551 | . . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V) | |
5 | 4 | ineq2i 3811 | . . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V)) |
6 | cnvin 5540 | . . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴})) | |
7 | df-res 5126 | . . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V)) | |
8 | 5, 6, 7 | 3eqtr4ri 2655 | . . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴})) |
9 | 8 | eqeq1i 2627 | . . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅) |
10 | inss2 3834 | . . . . . 6 ⊢ (𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴}) | |
11 | relxp 5227 | . . . . . 6 ⊢ Rel (V × {𝐴}) | |
12 | relss 5206 | . . . . . 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 5591 | . . . . 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 2846 | . 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 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 × cxp 5112 ◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |