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Theorem dfxrn2 37757
Description: Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)
Assertion
Ref Expression
dfxrn2 (𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
Distinct variable groups:   𝑢,𝑅,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Proof of Theorem dfxrn2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 xrnrel 37754 . . 3 Rel (𝑅𝑆)
2 dfrel4v 6182 . . 3 (Rel (𝑅𝑆) ↔ (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧})
31, 2mpbi 229 . 2 (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
4 breq2 5145 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅𝑆)𝑧𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩))
5 brxrn2 37756 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
65elv 3474 . . . . 5 (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
7 brxrn 37755 . . . . . . . . 9 ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦)))
87el3v 37597 . . . . . . . 8 (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦))
98anbi2i 622 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
10 3anass 1092 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
119, 10bitr4i 278 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
12112exbii 1843 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
134copsex2gb 5799 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
146, 12, 133bitr2i 299 . . . 4 (𝑢(𝑅𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
1514simplbi 497 . . 3 (𝑢(𝑅𝑆)𝑧𝑧 ∈ (V × V))
164, 15cnvoprab 8042 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
178oprabbii 7471 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
1817cnveqi 5867 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
193, 16, 183eqtr2i 2760 1 (𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  cop 4629   class class class wbr 5141  {copab 5203   × cxp 5667  ccnv 5668  Rel wrel 5674  {coprab 7405  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-sbc 3773  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-oprab 7408  df-1st 7971  df-2nd 7972  df-xrn 37752
This theorem is referenced by: (None)
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