Theorem rnxrn 36079

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 1093	. . . . 5 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
2	1	3exbii 1852	. . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
3		exrot3 2170	. . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
4		19.42v 1955	. . . . 5 ⊢ (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
5	4	2exbii 1851	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	2, 3, 5	3bitri 301	. . 3 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
7	6	abbii 2824	. 2 ⊢ {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))}
8		dfrn6 35993	. . 3 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅}
9		n0 4246	. . . . 5 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆))
10		elec1cnvxrn2 36078	. . . . . . 7 ⊢ (𝑢 ∈ V → (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
11	10	elv 3416	. . . . . 6 ⊢ (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
12	11	exbii 1850	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
13	9, 12	bitri 278	. . . 4 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
14	13	abbii 2824	. . 3 ⊢ {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
15	8, 14	eqtri 2782	. 2 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
16		df-opab 5096	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))}
17	7, 15, 16	3eqtr4i 2792	1 ⊢ ran (𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

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