Theorem supiso 8381

Description: Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supiso.1	|- ( ph -> F Isom R , S ( A , B ) )
supiso.2	|- ( ph -> C C_ A )
supisoex.3	|- ( ph -> E. x e. A ( A. y e. C -. x R y /\ A. y e. A ( y R x -> E. z e. C y R z ) ) )
supiso.4	|- ( ph -> R Or A )

Assertion

Ref	Expression
supiso	|- ( ph -> sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) )

Distinct variable groups:   x, y, z, A   x, C, y, z   x, F, y, z   x, R, y, z   x, S, y, z   x, B, y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem supiso

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supiso.4	. . 3 |- ( ph -> R Or A )
2		supiso.1	. . . 4 |- ( ph -> F Isom R , S ( A , B ) )
3		isoso 6598	. . . 4 |- ( F Isom R , S ( A , B ) -> ( R Or A <-> S Or B ) )
4	2, 3	syl 17	. . 3 |- ( ph -> ( R Or A <-> S Or B ) )
5	1, 4	mpbid 222	. 2 |- ( ph -> S Or B )
6		isof1o 6573	. . . 4 |- ( F Isom R , S ( A , B ) -> F : A -1-1-onto-> B )
7		f1of 6137	. . . 4 |- ( F : A -1-1-onto-> B -> F : A --> B )
8	2, 6, 7	3syl 18	. . 3 |- ( ph -> F : A --> B )
9		supisoex.3	. . . 4 |- ( ph -> E. x e. A ( A. y e. C -. x R y /\ A. y e. A ( y R x -> E. z e. C y R z ) ) )
10	1, 9	supcl 8364	. . 3 |- ( ph -> sup ( C , A , R ) e. A )
11	8, 10	ffvelrnd 6360	. 2 |- ( ph -> ( F ` sup ( C , A , R ) ) e. B )
12	1, 9	supub 8365	. . . . . 6 |- ( ph -> ( u e. C -> -. sup ( C , A , R ) R u ) )
13	12	ralrimiv 2965	. . . . 5 |- ( ph -> A. u e. C -. sup ( C , A , R ) R u )
14	1, 9	suplub 8366	. . . . . . 7 |- ( ph -> ( ( u e. A /\ u R sup ( C , A , R ) ) -> E. z e. C u R z ) )
15	14	expd 452	. . . . . 6 |- ( ph -> ( u e. A -> ( u R sup ( C , A , R ) -> E. z e. C u R z ) ) )
16	15	ralrimiv 2965	. . . . 5 |- ( ph -> A. u e. A ( u R sup ( C , A , R ) -> E. z e. C u R z ) )
17		supiso.2	. . . . . . 7 |- ( ph -> C C_ A )
18	2, 17	supisolem 8379	. . . . . 6 |- ( ( ph /\ sup ( C , A , R ) e. A ) -> ( ( A. u e. C -. sup ( C , A , R ) R u /\ A. u e. A ( u R sup ( C , A , R ) -> E. z e. C u R z ) ) <-> ( A. w e. ( F " C ) -. ( F ` sup ( C , A , R ) ) S w /\ A. w e. B ( w S ( F ` sup ( C , A , R ) ) -> E. v e. ( F " C ) w S v ) ) ) )
19	10, 18	mpdan 702	. . . . 5 |- ( ph -> ( ( A. u e. C -. sup ( C , A , R ) R u /\ A. u e. A ( u R sup ( C , A , R ) -> E. z e. C u R z ) ) <-> ( A. w e. ( F " C ) -. ( F ` sup ( C , A , R ) ) S w /\ A. w e. B ( w S ( F ` sup ( C , A , R ) ) -> E. v e. ( F " C ) w S v ) ) ) )
20	13, 16, 19	mpbi2and 956	. . . 4 |- ( ph -> ( A. w e. ( F " C ) -. ( F ` sup ( C , A , R ) ) S w /\ A. w e. B ( w S ( F ` sup ( C , A , R ) ) -> E. v e. ( F " C ) w S v ) ) )
21	20	simpld 475	. . 3 |- ( ph -> A. w e. ( F " C ) -. ( F ` sup ( C , A , R ) ) S w )
22	21	r19.21bi 2932	. 2 |- ( ( ph /\ w e. ( F " C ) ) -> -. ( F ` sup ( C , A , R ) ) S w )
23	20	simprd 479	. . . 4 |- ( ph -> A. w e. B ( w S ( F ` sup ( C , A , R ) ) -> E. v e. ( F " C ) w S v ) )
24	23	r19.21bi 2932	. . 3 |- ( ( ph /\ w e. B ) -> ( w S ( F ` sup ( C , A , R ) ) -> E. v e. ( F " C ) w S v ) )
25	24	impr 649	. 2 |- ( ( ph /\ ( w e. B /\ w S ( F ` sup ( C , A , R ) ) ) ) -> E. v e. ( F " C ) w S v )
26	5, 11, 22, 25	eqsupd 8363	1 |- ( ph -> sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   Or wor 5034   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   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

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-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-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-sup 8348

This theorem is referenced by:  infiso  8413  infrenegsup  11006