Theorem bdcuni 7250

Description: The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)

Assertion

Ref	Expression
bdcuni	⊢ BOUNDED ∪ x

Proof of Theorem bdcuni

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-bdel 7195	. . . . 5 ⊢ BOUNDED y ∈ z
2	1	ax-bdex 7193	. . . 4 ⊢ BOUNDED ∃z ∈ x y ∈ z
3	2	bdcab 7223	. . 3 ⊢ BOUNDED {y ∣ ∃z ∈ x y ∈ z}
4		df-rex 2290	. . . . 5 ⊢ (∃z ∈ x y ∈ z ↔ ∃z(z ∈ x ∧ y ∈ z))
5		exancom 1481	. . . . 5 ⊢ (∃z(z ∈ x ∧ y ∈ z) ↔ ∃z(y ∈ z ∧ z ∈ x))
6	4, 5	bitri 173	. . . 4 ⊢ (∃z ∈ x y ∈ z ↔ ∃z(y ∈ z ∧ z ∈ x))
7	6	abbii 2135	. . 3 ⊢ {y ∣ ∃z ∈ x y ∈ z} = {y ∣ ∃z(y ∈ z ∧ z ∈ x)}
8	3, 7	bdceqi 7217	. 2 ⊢ BOUNDED {y ∣ ∃z(y ∈ z ∧ z ∈ x)}
9		df-uni 3555	. 2 ⊢ ∪ x = {y ∣ ∃z(y ∈ z ∧ z ∈ x)}
10	8, 9	bdceqir 7218	1 ⊢ BOUNDED ∪ x

Syntax hints:   ∧ wa 97  ∃wex 1362  {cab 2008  ∃wrex 2285  ∪ cuni 3554  BOUNDED wbdc 7214

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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7187  ax-bdex 7193  ax-bdel 7195  ax-bdsb 7196

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-rex 2290  df-uni 3555  df-bdc 7215

This theorem is referenced by:  bj-uniex2  7286