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Theorem dminxp 4793
Description: Domain of the intersection with a cross 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 4555 . . . 4 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐴 × 𝐵))
2 cnvin 4759 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶(𝐴 × 𝐵))
3 cnvxp 4770 . . . . . . 7 (𝐴 × 𝐵) = (𝐵 × 𝐴)
43ineq2i 3163 . . . . . 6 (𝐶(𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
52, 4eqtri 2076 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
65rneqi 4590 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
71, 6eqtri 2076 . . 3 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
87eqeq1i 2063 . 2 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9 rninxp 4792 . 2 (ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥)
10 vex 2577 . . . . 5 𝑦 ∈ V
11 vex 2577 . . . . 5 𝑥 ∈ V
1210, 11brcnv 4546 . . . 4 (𝑦𝐶𝑥𝑥𝐶𝑦)
1312rexbii 2348 . . 3 (∃𝑦𝐵 𝑦𝐶𝑥 ↔ ∃𝑦𝐵 𝑥𝐶𝑦)
1413ralbii 2347 . 2 (∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
158, 9, 143bitri 199 1 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wral 2323  wrex 2324  cin 2944   class class class wbr 3792   × cxp 4371  ccnv 4372  dom cdm 4373  ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by: (None)
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