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Theorem dfxrn2 36152
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 36149 . . 3 Rel (𝑅𝑆)
2 dfrel4v 6023 . . 3 (Rel (𝑅𝑆) ↔ (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧})
31, 2mpbi 233 . 2 (𝑅𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
4 breq2 5035 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅𝑆)𝑧𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩))
5 brxrn2 36151 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
65elv 3405 . . . . 5 (𝑢(𝑅𝑆)𝑧 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
7 brxrn 36150 . . . . . . . . 9 ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦)))
87el3v 36017 . . . . . . . 8 (𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥𝑢𝑆𝑦))
98anbi2i 626 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
10 3anass 1096 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
119, 10bitr4i 281 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
12112exbii 1855 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
134copsex2gb 5651 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
146, 12, 133bitr2i 302 . . . 4 (𝑢(𝑅𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅𝑆)𝑧))
1514simplbi 501 . . 3 (𝑢(𝑅𝑆)𝑧𝑧 ∈ (V × V))
164, 15cnvoprab 7786 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅𝑆)𝑧}
178oprabbii 7238 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
1817cnveqi 5718 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
193, 16, 183eqtr2i 2768 1 (𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  Vcvv 3399  cop 4523   class class class wbr 5031  {copab 5093   × cxp 5524  ccnv 5525  Rel wrel 5531  {coprab 7174  cxrn 35978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348  df-oprab 7177  df-1st 7717  df-2nd 7718  df-xrn 36147
This theorem is referenced by: (None)
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