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Theorem rninxp 4950
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 3055 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ∀𝑦𝐵 𝑦 ∈ ran (𝐶𝐴))
2 ssrnres 4949 . 2 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3 df-ima 4520 . . . . 5 (𝐶𝐴) = ran (𝐶𝐴)
43eleq2i 2182 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ 𝑦 ∈ ran (𝐶𝐴))
5 vex 2661 . . . . 5 𝑦 ∈ V
65elima 4854 . . . 4 (𝑦 ∈ (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
74, 6bitr3i 185 . . 3 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝐴 𝑥𝐶𝑦)
87ralbii 2416 . 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 1314  wcel 1463  wral 2391  wrex 2392  cin 3038  wss 3039   class class class wbr 3897   × cxp 4505  ran crn 4508  cres 4509  cima 4510
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  dminxp  4951  fncnv  5157
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