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Theorem rninxp 6003
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 3903 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴))
2 ssrnres 6002 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3 df-ima 5532 . . . . 5 (𝐶𝐴) = ran (𝐶𝐴)
43eleq2i 2881 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ 𝑦 ∈ ran (𝐶𝐴))
5 vex 3444 . . . . 5 𝑦 ∈ V
65elima 5901 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
74, 6bitr3i 280 . . 3 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
87ralbii 3133 . 2 (∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴) ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
91, 2, 83bitr3i 304 1 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2111  wral 3106  wrex 3107  cin 3880  wss 3881   class class class wbr 5030   × cxp 5517  ran crn 5520  cres 5521  cima 5522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532
This theorem is referenced by:  dminxp  6004  fncnv  6397  exfo  6848  brdom3  9939  brdom5  9940  brdom4  9941
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