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Theorem dminxp 5493
 Description: Domain of the intersection 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 5238 . . . 4 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐴 × 𝐵))
2 cnvin 5459 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶(𝐴 × 𝐵))
3 cnvxp 5470 . . . . . . 7 (𝐴 × 𝐵) = (𝐵 × 𝐴)
43ineq2i 3773 . . . . . 6 (𝐶(𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
52, 4eqtri 2632 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
65rneqi 5273 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
71, 6eqtri 2632 . . 3 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
87eqeq1i 2615 . 2 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9 rninxp 5492 . 2 (ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥)
10 vex 3176 . . . . 5 𝑦 ∈ V
11 vex 3176 . . . . 5 𝑥 ∈ V
1210, 11brcnv 5227 . . . 4 (𝑦𝐶𝑥𝑥𝐶𝑦)
1312rexbii 3023 . . 3 (∃𝑦𝐵 𝑦𝐶𝑥 ↔ ∃𝑦𝐵 𝑥𝐶𝑦)
1413ralbii 2963 . 2 (∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
158, 9, 143bitri 285 1 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  trust  21843
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