Theorem uniexg 4316

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 4315	. 2 ⊢ ∪A = ⋃1(◡k Sk “k A)
2		ssetkex 4294	. . . . 5 ⊢ Sk ∈ V
3	2	cnvkex 4287	. . . 4 ⊢ ◡k Sk ∈ V
4		imakexg 4299	. . . 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 4292	. . 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 2859  ∪cuni 3891  ⋃₁cuni1 4133  ^◡_kccnvk 4175   “_k cimak 4179   S_k cssetk 4183

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193

This theorem is referenced by:  uniex  4317  pw1exb  4326