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Theorem dminxp 5732
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 5471 . . . 4 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐴 × 𝐵))
2 cnvin 5698 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶(𝐴 × 𝐵))
3 cnvxp 5709 . . . . . . 7 (𝐴 × 𝐵) = (𝐵 × 𝐴)
43ineq2i 3954 . . . . . 6 (𝐶(𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
52, 4eqtri 2782 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
65rneqi 5507 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
71, 6eqtri 2782 . . 3 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
87eqeq1i 2765 . 2 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9 rninxp 5731 . 2 (ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥)
10 vex 3343 . . . . 5 𝑦 ∈ V
11 vex 3343 . . . . 5 𝑥 ∈ V
1210, 11brcnv 5460 . . . 4 (𝑦𝐶𝑥𝑥𝐶𝑦)
1312rexbii 3179 . . 3 (∃𝑦𝐵 𝑦𝐶𝑥 ↔ ∃𝑦𝐵 𝑥𝐶𝑦)
1413ralbii 3118 . 2 (∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
158, 9, 143bitri 286 1 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wral 3050  wrex 3051  cin 3714   class class class wbr 4804   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267
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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
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-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  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-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279
This theorem is referenced by:  trust  22234
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