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Theorem reliin 4402
 Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
Assertion
Ref Expression
reliin (x A Rel B → Rel x A B)

Proof of Theorem reliin
StepHypRef Expression
1 iinss 3699 . 2 (x A B ⊆ (V × V) → x A B ⊆ (V × V))
2 df-rel 4295 . . 3 (Rel BB ⊆ (V × V))
32rexbii 2325 . 2 (x A Rel Bx A B ⊆ (V × V))
4 df-rel 4295 . 2 (Rel x A B x A B ⊆ (V × V))
51, 3, 43imtr4i 190 1 (x A Rel B → Rel x A B)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wrex 2301  Vcvv 2551   ⊆ wss 2911  ∩ ciin 3649   × cxp 4286  Rel wrel 4293 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iin 3651  df-rel 4295 This theorem is referenced by:  relint  4404  xpiindim  4416
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