Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnxrn Structured version   Visualization version   GIF version

Theorem rnxrn 34491
Description: Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
Assertion
Ref Expression
rnxrn ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
Distinct variable groups:   𝑢,𝑅,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Proof of Theorem rnxrn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 3anass 1079 . . . . 5 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
213exbii 1925 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
3 exrot3 2200 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
4 19.42v 2032 . . . . 5 (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
542exbii 1924 . . . 4 (∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
62, 3, 53bitri 286 . . 3 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
76abbii 2887 . 2 {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
8 dfrn6 34408 . . 3 ran (𝑅𝑆) = {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅}
9 n0 4076 . . . . 5 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆))
10 elec1cnvxrn2 34490 . . . . . . 7 (𝑢 ∈ V → (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
1110elv 34321 . . . . . 6 (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1211exbii 1923 . . . . 5 (∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
139, 12bitri 264 . . . 4 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1413abbii 2887 . . 3 {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
158, 14eqtri 2792 . 2 ran (𝑅𝑆) = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
16 df-opab 4845 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
177, 15, 163eqtr4i 2802 1 ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wne 2942  Vcvv 3349  c0 4061  cop 4320   class class class wbr 4784  {copab 4844  ccnv 5248  ran crn 5250  [cec 7893  cxrn 34307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7314  df-2nd 7315  df-ec 7897  df-xrn 34468
This theorem is referenced by:  rnxrnres  34492  dfcoss4  34508  dfssr2  34584
  Copyright terms: Public domain W3C validator