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Theorem rninxp 6178
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 3970 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴))
2 ssrnres 6177 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3 df-ima 5689 . . . . 5 (𝐶𝐴) = ran (𝐶𝐴)
43eleq2i 2825 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ 𝑦 ∈ ran (𝐶𝐴))
5 vex 3478 . . . . 5 𝑦 ∈ V
65elima 6064 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
74, 6bitr3i 276 . . 3 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
87ralbii 3093 . 2 (∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴) ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
91, 2, 83bitr3i 300 1 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cin 3947  wss 3948   class class class wbr 5148   × cxp 5674  ran crn 5677  cres 5678  cima 5679
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  dminxp  6179  fncnv  6621  exfo  7106  brdom3  10525  brdom5  10526  brdom4  10527
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