Theorem iunsnima 29428

Description: Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)

Hypotheses

Ref	Expression
iunsnima.1	|- ( ph -> A e. V )
iunsnima.2	|- ( ( ph /\ x e. A ) -> B e. W )

Assertion

Ref	Expression
iunsnima	|- ( ( ph /\ x e. A ) -> ( U_ x e. A ( { x } X. B ) " { x } ) = B )

Proof of Theorem iunsnima

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- x e. _V
2		vex 3203	. . . 4 |- y e. _V
3	1, 2	elimasn 5490	. . 3 |- ( y e. ( U_ x e. A ( { x } X. B ) " { x } ) <-> <. x , y >. e. U_ x e. A ( { x } X. B ) )
4		opeliunxp 5170	. . . . 5 |- ( <. x , y >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ y e. B ) )
5	4	baib 944	. . . 4 |- ( x e. A -> ( <. x , y >. e. U_ x e. A ( { x } X. B ) <-> y e. B ) )
6	5	adantl 482	. . 3 |- ( ( ph /\ x e. A ) -> ( <. x , y >. e. U_ x e. A ( { x } X. B ) <-> y e. B ) )
7	3, 6	syl5bb 272	. 2 |- ( ( ph /\ x e. A ) -> ( y e. ( U_ x e. A ( { x } X. B ) " { x } ) <-> y e. B ) )
8	7	eqrdv 2620	1 |- ( ( ph /\ x e. A ) -> ( U_ x e. A ( { x } X. B ) " { x } ) = B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   "cima 5117

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-eu 2474  df-mo 2475  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-sbc 3436  df-csb 3534  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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  esum2d  30155