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Theorem rnxrn 36079
 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 1093 . . . . 5 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
213exbii 1852 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
3 exrot3 2170 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
4 19.42v 1955 . . . . 5 (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
542exbii 1851 . . . 4 (∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
62, 3, 53bitri 301 . . 3 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
76abbii 2824 . 2 {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
8 dfrn6 35993 . . 3 ran (𝑅𝑆) = {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅}
9 n0 4246 . . . . 5 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆))
10 elec1cnvxrn2 36078 . . . . . . 7 (𝑢 ∈ V → (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
1110elv 3416 . . . . . 6 (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1211exbii 1850 . . . . 5 (∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
139, 12bitri 278 . . . 4 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1413abbii 2824 . . 3 {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
158, 14eqtri 2782 . 2 ran (𝑅𝑆) = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
16 df-opab 5096 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
177, 15, 163eqtr4i 2792 1 ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {cab 2736   ≠ wne 2952  Vcvv 3410  ∅c0 4226  ⟨cop 4529   class class class wbr 5033  {copab 5095  ◡ccnv 5524  ran crn 5526  [cec 8298   ⋉ cxrn 35885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fo 6342  df-fv 6344  df-1st 7694  df-2nd 7695  df-ec 8302  df-xrn 36056 This theorem is referenced by:  rnxrnres  36080  dfcoss4  36096  dfssr2  36172
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