ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relelfvdm Structured version   GIF version

Theorem relelfvdm 5148
Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
Assertion
Ref Expression
relelfvdm ((Rel 𝐹 A (𝐹B)) → B dom 𝐹)

Proof of Theorem relelfvdm
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfv 5119 . . . . . 6 (A (𝐹B) ↔ x(A x y(B𝐹yy = x)))
2 exsimpr 1506 . . . . . 6 (x(A x y(B𝐹yy = x)) → xy(B𝐹yy = x))
31, 2sylbi 114 . . . . 5 (A (𝐹B) → xy(B𝐹yy = x))
4 equsb1 1665 . . . . . . . 8 [x / y]y = x
5 spsbbi 1722 . . . . . . . 8 (y(B𝐹yy = x) → ([x / y]B𝐹y ↔ [x / y]y = x))
64, 5mpbiri 157 . . . . . . 7 (y(B𝐹yy = x) → [x / y]B𝐹y)
7 nfv 1418 . . . . . . . 8 y B𝐹x
8 breq2 3759 . . . . . . . 8 (y = x → (B𝐹yB𝐹x))
97, 8sbie 1671 . . . . . . 7 ([x / y]B𝐹yB𝐹x)
106, 9sylib 127 . . . . . 6 (y(B𝐹yy = x) → B𝐹x)
1110eximi 1488 . . . . 5 (xy(B𝐹yy = x) → x B𝐹x)
123, 11syl 14 . . . 4 (A (𝐹B) → x B𝐹x)
1312anim2i 324 . . 3 ((Rel 𝐹 A (𝐹B)) → (Rel 𝐹 x B𝐹x))
14 19.42v 1783 . . 3 (x(Rel 𝐹 B𝐹x) ↔ (Rel 𝐹 x B𝐹x))
1513, 14sylibr 137 . 2 ((Rel 𝐹 A (𝐹B)) → x(Rel 𝐹 B𝐹x))
16 releldm 4512 . . 3 ((Rel 𝐹 B𝐹x) → B dom 𝐹)
1716exlimiv 1486 . 2 (x(Rel 𝐹 B𝐹x) → B dom 𝐹)
1815, 17syl 14 1 ((Rel 𝐹 A (𝐹B)) → B dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378   wcel 1390  [wsb 1642   class class class wbr 3755  dom cdm 4288  Rel wrel 4293  cfv 4845
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-iota 4810  df-fv 4853
This theorem is referenced by:  elmpt2cl  5640  mpt2xopn0yelv  5795  eluzel2  8214
  Copyright terms: Public domain W3C validator