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Theorem elrn3 31353
Description: Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
Assertion
Ref Expression
elrn3 (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Proof of Theorem elrn3
StepHypRef Expression
1 df-rn 5090 . . 3 ran 𝐵 = dom 𝐵
21eleq2i 2696 . 2 (𝐴 ∈ ran 𝐵𝐴 ∈ dom 𝐵)
3 eldm3 31352 . 2 (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
4 cnvxp 5514 . . . . . . 7 (V × {𝐴}) = ({𝐴} × V)
54ineq2i 3794 . . . . . 6 (𝐵(V × {𝐴})) = (𝐵 ∩ ({𝐴} × V))
6 cnvin 5503 . . . . . 6 (𝐵 ∩ (V × {𝐴})) = (𝐵(V × {𝐴}))
7 df-res 5091 . . . . . 6 (𝐵 ↾ {𝐴}) = (𝐵 ∩ ({𝐴} × V))
85, 6, 73eqtr4ri 2659 . . . . 5 (𝐵 ↾ {𝐴}) = (𝐵 ∩ (V × {𝐴}))
98eqeq1i 2631 . . . 4 ((𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅)
10 inss2 3817 . . . . . 6 (𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴})
11 relxp 5193 . . . . . 6 Rel (V × {𝐴})
12 relss 5172 . . . . . 6 ((𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴}) → (Rel (V × {𝐴}) → Rel (𝐵 ∩ (V × {𝐴}))))
1310, 11, 12mp2 9 . . . . 5 Rel (𝐵 ∩ (V × {𝐴}))
14 cnveq0 5553 . . . . 5 (Rel (𝐵 ∩ (V × {𝐴})) → ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅))
1513, 14ax-mp 5 . . . 4 ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅)
169, 15bitr4i 267 . . 3 ((𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅)
1716necon3bii 2848 . 2 ((𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
182, 3, 173bitri 286 1 (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1992  wne 2796  Vcvv 3191  cin 3559  wss 3560  c0 3896  {csn 4153   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091
This theorem is referenced by: (None)
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