Theorem dfxrn2 38720

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 38717	. . 3 ⊢ Rel (𝑅 ⋉ 𝑆)
2		dfrel4v 6148	. . 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 5090	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
5		brxrn2 38719	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	5	elv 3435	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
7		brxrn 38718	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
8	7	el3v 3438	. . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
9	8	anbi2i 624	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
10		3anass 1095	. . . . . . 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 1851	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
13	4	copsex2gb 5755	. . . . 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 496	. . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V))
16	4, 15	cnvoprab 8006	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
17	8	oprabbii 7427	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
18	17	cnveqi 5823	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
19	3, 16, 18	3eqtr2i 2766	1 ⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   class class class wbr 5086  {copab 5148   × cxp 5622  ^◡ccnv 5623  Rel wrel 5629  {coprab 7361   ⋉ cxrn 38509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-oprab 7364  df-1st 7935  df-2nd 7936  df-xrn 38715

This theorem is referenced by:  dmxrn  38722