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Mirrors > Home > MPE Home > Th. List > elrn2g | Structured version Visualization version 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 4870 | . . . 4 ⊢ (𝑦 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐴〉) | |
2 | 1 | eleq1d 2819 | . . 3 ⊢ (𝑦 = 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
3 | 2 | exbidv 1925 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
4 | dfrn3 5884 | . 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵} | |
5 | 3, 4 | elab2g 3668 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 〈cop 4630 ran crn 5673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-cnv 5680 df-dm 5682 df-rn 5683 |
This theorem is referenced by: elrng 5886 elrn2 5887 fvrnressn 7146 fo2ndf 8094 |
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