Theorem bnj1400 30906

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1400.1	|- ( y e. A -> A. x y e. A )

Assertion

Ref	Expression
bnj1400	|- dom U. A = U_ x e. A dom x

Distinct variable groups:   y, A   x, y

Allowed substitution hint:   A( x)

Proof of Theorem bnj1400

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmuni 5334	. 2 |- dom U. A = U_ z e. A dom z
2		df-iun 4522	. . 3 |- U_ x e. A dom x = { y | E. x e. A y e. dom x }
3		df-iun 4522	. . . 4 |- U_ z e. A dom z = { y | E. z e. A y e. dom z }
4		bnj1400.1	. . . . . . 7 |- ( y e. A -> A. x y e. A )
5	4	nfcii 2755	. . . . . 6 |- F/_ x A
6		nfcv 2764	. . . . . 6 |- F/_ z A
7		nfv 1843	. . . . . 6 |- F/ z y e. dom x
8		nfv 1843	. . . . . 6 |- F/ x y e. dom z
9		dmeq 5324	. . . . . . 7 |- ( x = z -> dom x = dom z )
10	9	eleq2d 2687	. . . . . 6 |- ( x = z -> ( y e. dom x <-> y e. dom z ) )
11	5, 6, 7, 8, 10	cbvrexf 3166	. . . . 5 |- ( E. x e. A y e. dom x <-> E. z e. A y e. dom z )
12	11	abbii 2739	. . . 4 |- { y | E. x e. A y e. dom x } = { y | E. z e. A y e. dom z }
13	3, 12	eqtr4i 2647	. . 3 |- U_ z e. A dom z = { y | E. x e. A y e. dom x }
14	2, 13	eqtr4i 2647	. 2 |- U_ x e. A dom x = U_ z e. A dom z
15	1, 14	eqtr4i 2647	1 |- dom U. A = U_ x e. A dom x

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   U.cuni 4436   U_ciun 4520   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-dm 5124

This theorem is referenced by:  bnj1398  31102  bnj1450  31118  bnj1498  31129  bnj1501  31135