Theorem dfxrn2 38365

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 38362	. . 3 ⊢ Rel (𝑅 ⋉ 𝑆)
2		dfrel4v 6166	. . 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 5114	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
5		brxrn2 38364	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	5	elv 3455	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
7		brxrn 38363	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
8	7	el3v 3458	. . . . . . . 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 5772	. . . . 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 8042	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
17	8	oprabbii 7459	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
18	17	cnveqi 5841	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
19	3, 16, 18	3eqtr2i 2759	1 ⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   class class class wbr 5110  {copab 5172   × cxp 5639  ^◡ccnv 5640  Rel wrel 5646  {coprab 7391   ⋉ cxrn 38175

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-oprab 7394  df-1st 7971  df-2nd 7972  df-xrn 38360

This theorem is referenced by:  dmxrn  38367