Theorem ump 10459

Description: The union of a part of a powerset belongs to it.

Assertion

Ref	Expression
ump	|- (A e. B -> U.{x e. P~A | ph} e. P~A)

Distinct variable group:   x,A

Proof of Theorem ump

Step	Hyp	Ref	Expression
1		ssrab2 2131	. . 3 |- {x e. P~A | ph} (_ P~A
2		sspwuni 2758	. . 3 |- ({x e. P~A | ph} (_ P~A <-> U.{x e. P~A | ph} (_ A)
3	1, 2	mpbi 189	. 2 |- U.{x e. P~A | ph} (_ A
4		elpw2g 2727	. 2 |- (A e. B -> (U.{x e. P~A | ph} e. P~A <-> U.{x e. P~A | ph} (_ A))
5	3, 4	mpbiri 194	1 |- (A e. B -> U.{x e. P~A | ph} e. P~A)

Syntax hints:   -> wi 3   e. wcel 958  {crab 1648   (_ wss 2047  P~cpw 2401  U.cuni 2503

This theorem is referenced by:  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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