Theorem dfxrn2 38394

Description: Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)

Assertion

Ref	Expression
dfxrn2	⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

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

Proof of Theorem dfxrn2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrnrel 38391	. . 3 ⊢ Rel (𝑅 ⋉ 𝑆)
2		dfrel4v 6179	. . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧})
3	1, 2	mpbi 230	. 2 ⊢ (𝑅 ⋉ 𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
4		breq2 5123	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
5		brxrn2 38393	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	5	elv 3464	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
7		brxrn 38392	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
8	7	el3v 3467	. . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
9	8	anbi2i 623	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
10		3anass 1094	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
11	9, 10	bitr4i 278	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
12	11	2exbii 1849	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
13	4	copsex2gb 5785	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧))
14	6, 12, 13	3bitr2i 299	. . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧))
15	14	simplbi 497	. . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V))
16	4, 15	cnvoprab 8059	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
17	8	oprabbii 7474	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
18	17	cnveqi 5854	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
19	3, 16, 18	3eqtr2i 2764	1 ⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   class class class wbr 5119  {copab 5181   × cxp 5652  ^◡ccnv 5653  Rel wrel 5659  {coprab 7406   ⋉ cxrn 38198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-oprab 7409  df-1st 7988  df-2nd 7989  df-xrn 38389

This theorem is referenced by: (None)