Theorem idjusnfi 14270

Description: The image of a singleton through an indexed disjoint union, where it is of finite sets. (Contributed by Jim Kingdon, 2-Oct-2026.)

Hypotheses

Ref	Expression
idjusnfi.a	|- ( ph -> A e. V )
idjusnfi.r	|- ( ( ph /\ j e. A ) -> C e. Fin )
idjusnfi.u	|- U = U_ j e. A ( { j } X. C )

Assertion

Ref	Expression
idjusnfi	|- ( ( ph /\ K e. A ) -> ( U " { K } ) e. Fin )

Distinct variable groups:   A, j   j, K   ph, j

Allowed substitution hints:   C( j)   U( j)   V( j)

Proof of Theorem idjusnfi

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idjusnfi.r	. . . 4 |- ( ( ph /\ j e. A ) -> C e. Fin )
2	1	ralrimiva 2623	. . 3 |- ( ph -> A. j e. A C e. Fin )
3		nfcsb1v 3180	. . . . 5 |- F/_ j [_ K / j ]_ C
4	3	nfel1 2403	. . . 4 |- F/ j [_ K / j ]_ C e. Fin
5		csbeq1a 3156	. . . . 5 |- ( j = K -> C = [_ K / j ]_ C )
6	5	eleq1d 2307	. . . 4 |- ( j = K -> ( C e. Fin <-> [_ K / j ]_ C e. Fin ) )
7	4, 6	rspc 2923	. . 3 |- ( K e. A -> ( A. j e. A C e. Fin -> [_ K / j ]_ C e. Fin ) )
8	2, 7	mpan9 281	. 2 |- ( ( ph /\ K e. A ) -> [_ K / j ]_ C e. Fin )
9		elimasng 5158	. . . . . . . 8 |- ( ( K e. A /\ q e. _V ) -> ( q e. ( U " { K } ) <-> <. K , q >. e. U ) )
10	9	elvd 2826	. . . . . . 7 |- ( K e. A -> ( q e. ( U " { K } ) <-> <. K , q >. e. U ) )
11		idjusnfi.u	. . . . . . . . 9 |- U = U_ j e. A ( { j } X. C )
12	11	eleq2i 2305	. . . . . . . 8 |- ( <. K , q >. e. U <-> <. K , q >. e. U_ j e. A ( { j } X. C ) )
13	3, 5	opeliunxp2f 6513	. . . . . . . 8 |- ( <. K , q >. e. U_ j e. A ( { j } X. C ) <-> ( K e. A /\ q e. [_ K / j ]_ C ) )
14	12, 13	bitri 184	. . . . . . 7 |- ( <. K , q >. e. U <-> ( K e. A /\ q e. [_ K / j ]_ C ) )
15	10, 14	bitrdi 196	. . . . . 6 |- ( K e. A -> ( q e. ( U " { K } ) <-> ( K e. A /\ q e. [_ K / j ]_ C ) ) )
16	15	bianabs 619	. . . . 5 |- ( K e. A -> ( q e. ( U " { K } ) <-> q e. [_ K / j ]_ C ) )
17	16	eqrdv 2236	. . . 4 |- ( K e. A -> ( U " { K } ) = [_ K / j ]_ C )
18	17	eleq1d 2307	. . 3 |- ( K e. A -> ( ( U " { K } ) e. Fin <-> [_ K / j ]_ C e. Fin ) )
19	18	adantl 277	. 2 |- ( ( ph /\ K e. A ) -> ( ( U " { K } ) e. Fin <-> [_ K / j ]_ C e. Fin ) )
20	8, 19	mpbird 167	1 |- ( ( ph /\ K e. A ) -> ( U " { K } ) e. Fin )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   [_csb 3147   {csn 3709   <.cop 3712   U_ciun 4012   X. cxp 4774   "cima 4779   Fincfn 7026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-iun 4014  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789

This theorem is used by:  gsum2d2fi  14274