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Theorem dfxrn2 35610
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 35607 . . 3 Rel (𝑅𝑆)
2 dfrel4v 6040 . . 3 (Rel (𝑅𝑆) ↔ (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧})
31, 2mpbi 232 . 2 (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
4 breq2 5061 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅𝑆)𝑧𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩))
5 brxrn2 35609 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
65elv 3498 . . . . 5 (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
7 brxrn 35608 . . . . . . . . 9 ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦)))
87el3v 35473 . . . . . . . 8 (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦))
98anbi2i 624 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
10 3anass 1089 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
119, 10bitr4i 280 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
12112exbii 1842 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
134copsex2gb 5672 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
146, 12, 133bitr2i 301 . . . 4 (𝑢(𝑅𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
1514simplbi 500 . . 3 (𝑢(𝑅𝑆)𝑧𝑧 ∈ (V × V))
164, 15cnvoprab 7750 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
178oprabbii 7213 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
1817cnveqi 5738 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
193, 16, 183eqtr2i 2848 1 (𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1081   = wceq 1530  wex 1773  wcel 2107  Vcvv 3493  cop 4565   class class class wbr 5057  {copab 5119   × cxp 5546  ccnv 5547  Rel wrel 5553  {coprab 7149  cxrn 35434
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-oprab 7152  df-1st 7681  df-2nd 7682  df-xrn 35605
This theorem is referenced by: (None)
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