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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 4403 | . . . 4 ⊢ (𝑦 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐴〉) | |
3 | 2 | eleq1d 2686 | . . 3 ⊢ (𝑦 = 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
4 | 3 | exbidv 1850 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) |
5 | dfrn3 5312 | . 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐵} | |
6 | 1, 4, 5 | elab2 3354 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ran crn 5115 |
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-rab 2921 df-v 3202 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-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: elrn 5366 dmrnssfld 5384 rniun 5543 ssrnres 5572 relssdmrn 5656 fvelrn 6352 tz7.48-1 7538 prsrn 29961 dfrn5 31677 |
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