Theorem fisupclrnmpt 39622

Description: A nonempty finite indexed set contains its supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fisupclrnmpt.x	|- F/ x ph
fisupclrnmpt.r	|- ( ph -> R Or A )
fisupclrnmpt.b	|- ( ph -> B e. Fin )
fisupclrnmpt.n	|- ( ph -> B =/= (/) )
fisupclrnmpt.c	|- ( ( ph /\ x e. B ) -> C e. A )

Assertion

Ref	Expression
fisupclrnmpt	|- ( ph -> sup ( ran ( x e. B |-> C ) , A , R ) e. A )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)   R( x)

Proof of Theorem fisupclrnmpt

Step	Hyp	Ref	Expression
1		fisupclrnmpt.x	. . 3 |- F/ x ph
2		eqid 2622	. . 3 |- ( x e. B |-> C ) = ( x e. B |-> C )
3		fisupclrnmpt.c	. . 3 |- ( ( ph /\ x e. B ) -> C e. A )
4	1, 2, 3	rnmptssd 39385	. 2 |- ( ph -> ran ( x e. B |-> C ) C_ A )
5		fisupclrnmpt.r	. . 3 |- ( ph -> R Or A )
6		fisupclrnmpt.b	. . . 4 |- ( ph -> B e. Fin )
7	2	rnmptfi 39351	. . . 4 |- ( B e. Fin -> ran ( x e. B |-> C ) e. Fin )
8	6, 7	syl 17	. . 3 |- ( ph -> ran ( x e. B |-> C ) e. Fin )
9		fisupclrnmpt.n	. . . 4 |- ( ph -> B =/= (/) )
10	1, 3, 2, 9	rnmptn0 39413	. . 3 |- ( ph -> ran ( x e. B |-> C ) =/= (/) )
11		fisupcl 8375	. . 3 |- ( ( R Or A /\ ( ran ( x e. B |-> C ) e. Fin /\ ran ( x e. B |-> C ) =/= (/) /\ ran ( x e. B |-> C ) C_ A ) ) -> sup ( ran ( x e. B |-> C ) , A , R ) e. ran ( x e. B |-> C ) )
12	5, 8, 10, 4, 11	syl13anc 1328	. 2 |- ( ph -> sup ( ran ( x e. B |-> C ) , A , R ) e. ran ( x e. B |-> C ) )
13	4, 12	sseldd 3604	1 |- ( ph -> sup ( ran ( x e. B |-> C ) , A , R ) e. A )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   Or wor 5034   ran crn 5115   Fincfn 7955   supcsup 8346

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-sup 8348

This theorem is referenced by:  uzublem  39657  limsupubuzlem  39944