Theorem xpiun 41766

Description: A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
xpiun	|- ( B X. C ) = U_ x e. B { <. a , b >. | ( a = x /\ b e. C ) }

Distinct variable groups:   x, B   C, a, b, x

Allowed substitution hints:   B( a, b)

Proof of Theorem xpiun

Step	Hyp	Ref	Expression
1		xpsnopab 41765	. . . . 5 |- ( { x } X. C ) = { <. a , b >. | ( a = x /\ b e. C ) }
2	1	eqcomi 2631	. . . 4 |- { <. a , b >. | ( a = x /\ b e. C ) } = ( { x } X. C )
3	2	a1i 11	. . 3 |- ( x e. B -> { <. a , b >. | ( a = x /\ b e. C ) } = ( { x } X. C ) )
4	3	iuneq2i 4539	. 2 |- U_ x e. B { <. a , b >. | ( a = x /\ b e. C ) } = U_ x e. B ( { x } X. C )
5		iunxpconst 5175	. 2 |- U_ x e. B ( { x } X. C ) = ( B X. C )
6	4, 5	eqtr2i 2645	1 |- ( B X. C ) = U_ x e. B { <. a , b >. | ( a = x /\ b e. C ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   U_ciun 4520   {copab 4712   X. cxp 5112

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-iun 4522  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)