Theorem rnxrn 38380

Description: Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)

Assertion

Ref	Expression
rnxrn	⊢ ran (𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Distinct variable groups:   𝑢,𝑅,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Proof of Theorem rnxrn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anass 1094	. . . . 5 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
2	1	3exbii 1847	. . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
3		exrot3 2163	. . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
4		19.42v 1951	. . . . 5 ⊢ (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
5	4	2exbii 1846	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	2, 3, 5	3bitri 297	. . 3 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
7	6	abbii 2807	. 2 ⊢ {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))}
8		dfrn6 38284	. . 3 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅}
9		n0 4359	. . . . 5 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆))
10		elec1cnvxrn2 38379	. . . . . . 7 ⊢ (𝑢 ∈ V → (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
11	10	elv 3483	. . . . . 6 ⊢ (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
12	11	exbii 1845	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
13	9, 12	bitri 275	. . . 4 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
14	13	abbii 2807	. . 3 ⊢ {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
15	8, 14	eqtri 2763	. 2 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
16		df-opab 5211	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))}
17	7, 15, 16	3eqtr4i 2773	1 ⊢ ran (𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938  Vcvv 3478  ∅c0 4339  ⟨cop 4637   class class class wbr 5148  {copab 5210  ^◡ccnv 5688  ran crn 5690  [cec 8742   ⋉ cxrn 38161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-ec 8746  df-xrn 38353

This theorem is referenced by:  rnxrnres  38381  dfcoss4  38397  dfssr2  38481