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Theorem rninxp 6210
Description: Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rninxp (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rninxp
StepHypRef Expression
1 dfss3 3997 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴))
2 ssrnres 6209 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3 df-ima 5713 . . . . 5 (𝐶𝐴) = ran (𝐶𝐴)
43eleq2i 2836 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ 𝑦 ∈ ran (𝐶𝐴))
5 vex 3492 . . . . 5 𝑦 ∈ V
65elima 6094 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
74, 6bitr3i 277 . . 3 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
87ralbii 3099 . 2 (∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴) ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
91, 2, 83bitr3i 301 1 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2108  wral 3067  wrex 3076  cin 3975  wss 3976   class class class wbr 5166   × cxp 5698  ran crn 5701  cres 5702  cima 5703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713
This theorem is referenced by:  dminxp  6211  fncnv  6651  exfo  7139  brdom3  10597  brdom5  10598  brdom4  10599
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