Theorem uniexg 4317

Description: The sum class of a set is a set. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
uniexg	⊢ (A ∈ V → ∪A ∈ V)

Proof of Theorem uniexg

Step	Hyp	Ref	Expression
1		dfuni3 4316	. 2 ⊢ ∪A = ⋃1(◡k Sk “k A)
2		ssetkex 4295	. . . . 5 ⊢ Sk ∈ V
3	2	cnvkex 4288	. . . 4 ⊢ ◡k Sk ∈ V
4		imakexg 4300	. . . 4 ⊢ ((◡k Sk ∈ V ∧ A ∈ V) → (◡k Sk “k A) ∈ V)
5	3, 4	mpan 651	. . 3 ⊢ (A ∈ V → (◡k Sk “k A) ∈ V)
6		uni1exg 4293	. . 3 ⊢ ((◡k Sk “k A) ∈ V → ⋃1(◡k Sk “k A) ∈ V)
7	5, 6	syl 15	. 2 ⊢ (A ∈ V → ⋃1(◡k Sk “k A) ∈ V)
8	1, 7	syl5eqel 2437	1 ⊢ (A ∈ V → ∪A ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1710  Vcvv 2860  ∪cuni 3892  ⋃₁cuni1 4134  ^◡_kccnvk 4176   “_k cimak 4180   S_k cssetk 4184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-opk 4059  df-1c 4137  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194

This theorem is referenced by:  uniex  4318  pw1exb  4327