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Theorem dminxp 5830
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 5563 . . . 4 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐴 × 𝐵))
2 cnvin 5796 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶(𝐴 × 𝐵))
3 cnvxp 5807 . . . . . . 7 (𝐴 × 𝐵) = (𝐵 × 𝐴)
43ineq2i 4034 . . . . . 6 (𝐶(𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
52, 4eqtri 2802 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) = (𝐶 ∩ (𝐵 × 𝐴))
65rneqi 5599 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
71, 6eqtri 2802 . . 3 dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (𝐶 ∩ (𝐵 × 𝐴))
87eqeq1i 2783 . 2 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9 rninxp 5829 . 2 (ran (𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥)
10 vex 3401 . . . . 5 𝑦 ∈ V
11 vex 3401 . . . . 5 𝑥 ∈ V
1210, 11brcnv 5552 . . . 4 (𝑦𝐶𝑥𝑥𝐶𝑦)
1312rexbii 3224 . . 3 (∃𝑦𝐵 𝑦𝐶𝑥 ↔ ∃𝑦𝐵 𝑥𝐶𝑦)
1413ralbii 3162 . 2 (∀𝑥𝐴𝑦𝐵 𝑦𝐶𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
158, 9, 143bitri 289 1 (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  wral 3090  wrex 3091  cin 3791   class class class wbr 4888   × cxp 5355  ccnv 5356  dom cdm 5357  ran crn 5358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370
This theorem is referenced by:  trust  22452
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