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Theorem dfxrn2 37709
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 37706 . . 3 Rel (𝑅𝑆)
2 dfrel4v 6189 . . 3 (Rel (𝑅𝑆) ↔ (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧})
31, 2mpbi 229 . 2 (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
4 breq2 5152 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅𝑆)𝑧𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩))
5 brxrn2 37708 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
65elv 3479 . . . . 5 (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
7 brxrn 37707 . . . . . . . . 9 ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦)))
87el3v 37549 . . . . . . . 8 (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦))
98anbi2i 622 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
10 3anass 1094 . . . . . . 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 1850 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
134copsex2gb 5806 . . . . 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 8050 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
178oprabbii 7479 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
1817cnveqi 5874 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
193, 16, 183eqtr2i 2765 1 (𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1086   = wceq 1540  wex 1780  wcel 2105  Vcvv 3473  cop 4634   class class class wbr 5148  {copab 5210   × cxp 5674  ccnv 5675  Rel wrel 5681  {coprab 7413  cxrn 37505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-oprab 7416  df-1st 7979  df-2nd 7980  df-xrn 37704
This theorem is referenced by: (None)
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