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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 4626 | . . . 4 ⊢ (𝑦 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐴〉) | |
3 | 2 | eleq1d 2891 | . . 3 ⊢ (𝑦 = 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
4 | 3 | exbidv 2020 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
5 | dfrn3 5548 | . 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵} | |
6 | 1, 4, 5 | elab2 3575 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1656 ∃wex 1878 ∈ wcel 2164 Vcvv 3414 〈cop 4405 ran crn 5347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-cnv 5354 df-dm 5356 df-rn 5357 |
This theorem is referenced by: elrn 5603 dmrnssfld 5621 rniun 5788 ssrnres 5817 relssdmrn 5901 fvelrn 6606 tz7.48-1 7809 prsrn 30502 dfrn5 32210 funressndmafv2rn 42119 |
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