Theorem uniexg 4470

Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A e. V instead of A e. _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant _V is one possible substitution for class variable V. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
uniexg	|- ( A e. V -> U. A e. _V )

Proof of Theorem uniexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3844	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2262	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2763	. . 3 |- x e. _V
4	3	uniex 4468	. 2 |- U. x e. _V
5	2, 4	vtoclg 2820	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   U.cuni 3835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-uni 3836

This theorem is referenced by:  uniexd  4471  abnexg  4477  snnex  4479  uniexb  4504  ssonuni  4520  dmexg  4926  rnexg  4927  elxp4  5153  elxp5  5154  iotaexab  5233  relrnfvex  5572  fvexg  5573  sefvex  5575  riotaexg  5877  iunexg  6171  1stvalg  6195  2ndvalg  6196  cnvf1o  6278  brtpos2  6304  tfrlemiex  6384  tfr1onlemex  6400  tfrcllemex  6413  en1bg  6854  en1uniel  6858  fival  7029  suplocexprlem2b  7774  suplocexprlemlub  7784  wrdexb  10926  restid  12861  tgval  12873  tgvalex  12874  istopon  14181  eltg  14220  eltg2  14221  tgss2  14247  ntrval  14278  restin  14344  cnovex  14364  cnprcl2k  14374  cnptopresti  14406  cnptoprest  14407  cnptoprest2  14408  lmtopcnp  14418  txbasex  14425  uptx  14442  reldvg  14833