![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elrn2g | GIF version |
Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
elrn2g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3777 | . . . 4 ⊢ (𝑦 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐴〉) | |
2 | 1 | eleq1d 2246 | . . 3 ⊢ (𝑦 = 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
3 | 2 | exbidv 1825 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
4 | dfrn3 4811 | . 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵} | |
5 | 3, 4 | elab2g 2884 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 〈cop 3594 ran crn 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-cnv 4630 df-dm 4632 df-rn 4633 |
This theorem is referenced by: elrng 4813 fvelrn 5642 fo2ndf 6221 |
Copyright terms: Public domain | W3C validator |