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Theorem dfxrn2 34479
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 34476 . . 3 Rel (𝑅𝑆)
2 dfrel4v 5742 . . 3 (Rel (𝑅𝑆) ↔ (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧})
31, 2mpbi 220 . 2 (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
4 breq2 4808 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅𝑆)𝑧𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩))
5 brxrn2 34478 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
65elv 34327 . . . . 5 (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
7 brxrn 34477 . . . . . . . . 9 ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦)))
87el3v 34331 . . . . . . . 8 (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦))
98anbi2i 732 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
10 3anass 1081 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
119, 10bitr4i 267 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
12112exbii 1924 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
134copsex2gb 5386 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
146, 12, 133bitr2i 288 . . . 4 (𝑢(𝑅𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
1514simplbi 478 . . 3 (𝑢(𝑅𝑆)𝑧𝑧 ∈ (V × V))
164, 15cnvoprab 7398 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
178oprabbii 6876 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
1817cnveqi 5452 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
193, 16, 183eqtr2i 2788 1 (𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cop 4327   class class class wbr 4804  {copab 4864   × cxp 5264  ccnv 5265  Rel wrel 5271  {coprab 6815  cxrn 34313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-oprab 6818  df-1st 7334  df-2nd 7335  df-xrn 34474
This theorem is referenced by: (None)
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