Theorem brxrn2 37756

Description: A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)

Assertion

Ref	Expression
brxrn2	⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑉,𝑦

Proof of Theorem brxrn2

Step	Hyp	Ref	Expression
1		xrnss3v 37753	. . . . . . 7 ⊢ (𝑅 ⋉ 𝑆) ⊆ (V × (V × V))
2	1	brel 5734	. . . . . 6 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
3	2	simprd 495	. . . . 5 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → 𝐵 ∈ (V × V))
4		elvv 5743	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 217	. . . 4 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
6	5	pm4.71ri 560	. . 3 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵))
7		19.41vv 1946	. . 3 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵))
8		breq2 5145	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
9	8	pm5.32i 574	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
10	9	2exbii 1843	. . 3 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)𝐵) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
11	6, 7, 10	3bitr2i 299	. 2 ⊢ (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
12		brxrn 37755	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
13	12	el3v23 37603	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
14	13	anbi2d 628	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))))
15		3anass 1092	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
16	14, 15	bitr4di 289	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
17	16	2exbidv 1919	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
18	11, 17	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629   class class class wbr 5141   × cxp 5667   ⋉ cxrn 37553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7971  df-2nd 7972  df-xrn 37752

This theorem is referenced by:  dfxrn2  37757  elecxrn  37767  inxpxrn  37776  br1cnvxrn2  37777