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Theorem rninxp 5030
Description: Range of the intersection with a cross 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 3118 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴))
2 ssrnres 5029 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3 df-ima 4600 . . . . 5 (𝐶𝐴) = ran (𝐶𝐴)
43eleq2i 2224 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ 𝑦 ∈ ran (𝐶𝐴))
5 vex 2715 . . . . 5 𝑦 ∈ V
65elima 4934 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
74, 6bitr3i 185 . . 3 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
87ralbii 2463 . 2 (∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴) ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
91, 2, 83bitr3i 209 1 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1335  wcel 2128  wral 2435  wrex 2436  cin 3101  wss 3102   class class class wbr 3966   × cxp 4585  ran crn 4588  cres 4589  cima 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-rel 4594  df-cnv 4595  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600
This theorem is referenced by:  dminxp  5031  fncnv  5237
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