Theorem rnxrn 39619

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 1111	. . . . 5 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
2	1	3exbii 1883	. . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
3		exrot3 2205	. . . 4 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
4		19.42v 1988	. . . . 5 ⊢ (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
5	4	2exbii 1882	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	2, 3, 5	3bitri 300	. . 3 ⊢ (∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
7	6	abbii 2830	. 2 ⊢ {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))}
8		dfrn6 39506	. . 3 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅}
9		n0 4303	. . . . 5 ⊢ ([𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆))
10		elec1cnvxrn2 39618	. . . . . . 7 ⊢ (𝑢 ∈ V → (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
11	10	elv 3459	. . . . . 6 ⊢ (𝑢 ∈ [𝑤]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
12	11	exbii 1881	. . . . 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 2830	. . 3 ⊢ {𝑤 ∣ [𝑤]◡(𝑅 ⋉ 𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
15	8, 14	eqtri 2786	. 2 ⊢ ran (𝑅 ⋉ 𝑆) = {𝑤 ∣ ∃𝑢∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
16		df-opab 5174	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))}
17	7, 15, 16	3eqtr4i 2796	1 ⊢ ran (𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ⟨cop 4840   class class class wbr 5108  {copab 5173  ^◡ccnv 5657  ran crn 5659  [cec 8753   ⋉ cxrn 39372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-1st 8035  df-2nd 8036  df-ec 8757  df-xrn 39578

This theorem is used by:  rnxrnres  39620  dfcoss4  39703  dfssr2  39777