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Theorem dminxp 6200
Description: Two ways to express totality of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
dminxp (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dminxp
StepHypRef Expression
1 dfdm4 5906 . . . 4 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐴 × 𝐵))
2 cnvin 6164 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶(𝐴 × 𝐵))
3 cnvxp 6177 . . . . . . 7 (𝐴 × 𝐵) = (𝐵 × 𝐴)
43ineq2i 4217 . . . . . 6 (𝐶(𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
52, 4eqtri 2765 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
65rneqi 5948 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
71, 6eqtri 2765 . . 3 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
87eqeq1i 2742 . 2 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9 rninxp 6199 . 2 (ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥)
10 vex 3484 . . . . 5 𝑦 ∈ V
11 vex 3484 . . . . 5 𝑥 ∈ V
1210, 11brcnv 5893 . . . 4 (𝑦𝐶𝑥𝑥𝐶𝑦)
1312rexbii 3094 . . 3 (∃𝑦𝐵 𝑦𝐶𝑥 ↔ ∃𝑦𝐵 𝑥𝐶𝑦)
1413ralbii 3093 . 2 (∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
158, 9, 143bitri 297 1 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wral 3061  wrex 3070  cin 3950   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  trust  24238  onsupmaxb  43251
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