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Mirrors > Home > MPE Home > Th. List > elrn2 | Structured version Visualization version GIF version |
Description: Membership in a range. (Contributed by NM, 10-Jul-1994.) |
Ref | Expression |
---|---|
elrn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elrn2 | ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opeq2 4797 | . . . 4 ⊢ (𝑦 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐴〉) | |
3 | 2 | eleq1d 2897 | . . 3 ⊢ (𝑦 = 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
4 | 3 | exbidv 1918 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
5 | dfrn3 5754 | . 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵} | |
6 | 1, 4, 5 | elab2 3669 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 〈cop 4566 ran crn 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 |
This theorem is referenced by: elrn 5816 dmrnssfld 5835 rniun 6000 ssrnres 6029 relssdmrn 6115 fvelrn 6838 tz7.48-1 8073 prsrn 31153 dfrn5 33012 funressndmafv2rn 43416 |
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