Theorem isomin 4876

Description: Isomorphisms preserve minimal elements. Note that (`'R"{D}) is Takeuti and Zaring's idiom for the initial segment {x | xRD}. Proposition 6.31(1) of [TakeutiZaring] p. 33.

Assertion

Ref	Expression
isomin	|- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> ((C i^i (`'R"{D})) = (/) <-> ((H"C) i^i (`'S"{(H` D)})) = (/)))

Proof of Theorem isomin

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . . . . . . . . . 14 |- (C C_ A -> (x e. C -> x e. A))
2		isof1o 4870	. . . . . . . . . . . . . . 15 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
3		f1ofn 4636	. . . . . . . . . . . . . . 15 |- (H:A-1-1-onto->B -> H Fn A)
4		visset 2295	. . . . . . . . . . . . . . . . 17 |- y e. _V
5	4	fnbrfvb 4712	. . . . . . . . . . . . . . . 16 |- ((H Fn A /\ x e. A) -> ((H` x) = y <-> xHy))
6	5	ex 402	. . . . . . . . . . . . . . 15 |- (H Fn A -> (x e. A -> ((H` x) = y <-> xHy)))
7	2, 3, 6	3syl 24	. . . . . . . . . . . . . 14 |- (H Isom R, S (A, B) -> (x e. A -> ((H` x) = y <-> xHy)))
8	1, 7	syl9r 72	. . . . . . . . . . . . 13 |- (H Isom R, S (A, B) -> (C C_ A -> (x e. C -> ((H` x) = y <-> xHy))))
9	8	imp31 389	. . . . . . . . . . . 12 |- (((H Isom R, S (A, B) /\ C C_ A) /\ x e. C) -> ((H` x) = y <-> xHy))
10	9	rexbidva 2120	. . . . . . . . . . 11 |- ((H Isom R, S (A, B) /\ C C_ A) -> (E.x e. C (H` x) = y <-> E.x e. C xHy))
11	4	elima 4270	. . . . . . . . . . 11 |- (y e. (H"C) <-> E.x e. C xHy)
12	10, 11	syl6rbbr 598	. . . . . . . . . 10 |- ((H Isom R, S (A, B) /\ C C_ A) -> (y e. (H"C) <-> E.x e. C (H` x) = y))
13		fvex 4689	. . . . . . . . . . . 12 |- (H` D) e. _V
14	4	eliniseg 4294	. . . . . . . . . . . 12 |- ((H` D) e. _V -> (y e. (`'S"{(H` D)}) <-> yS(H` D)))
15	13, 14	ax-mp 7	. . . . . . . . . . 11 |- (y e. (`'S"{(H` D)}) <-> yS(H` D))
16	15	a1i 8	. . . . . . . . . 10 |- ((H Isom R, S (A, B) /\ C C_ A) -> (y e. (`'S"{(H` D)}) <-> yS(H` D)))
17	12, 16	anbi12d 690	. . . . . . . . 9 |- ((H Isom R, S (A, B) /\ C C_ A) -> ((y e. (H"C) /\ y e. (`'S"{(H` D)})) <-> (E.x e. C (H` x) = y /\ yS(H` D))))
18		elin 2786	. . . . . . . . 9 |- (y e. ((H"C) i^i (`'S"{(H` D)})) <-> (y e. (H"C) /\ y e. (`'S"{(H` D)})))
19		r19.41v 2236	. . . . . . . . 9 |- (E.x e. C ((H` x) = y /\ yS(H` D)) <-> (E.x e. C (H` x) = y /\ yS(H` D)))
20	17, 18, 19	3bitr4g 614	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ C C_ A) -> (y e. ((H"C) i^i (`'S"{(H` D)})) <-> E.x e. C ((H` x) = y /\ yS(H` D))))
21	20	adantrr 431	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (y e. ((H"C) i^i (`'S"{(H` D)})) <-> E.x e. C ((H` x) = y /\ yS(H` D))))
22		visset 2295	. . . . . . . . . . . . . . . 16 |- x e. _V
23	22	eliniseg 4294	. . . . . . . . . . . . . . 15 |- (D e. A -> (x e. (`'R"{D}) <-> xRD))
24	23	ad2antll 443	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (x e. (`'R"{D}) <-> xRD))
25		isorel 4871	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (xRD <-> (H` x)S(H` D)))
26	24, 25	bitrd 587	. . . . . . . . . . . . 13 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (x e. (`'R"{D}) <-> (H` x)S(H` D)))
27		breq1 3341	. . . . . . . . . . . . . 14 |- ((H` x) = y -> ((H` x)S(H` D) <-> yS(H` D)))
28	27	biimpar 461	. . . . . . . . . . . . 13 |- (((H` x) = y /\ yS(H` D)) -> (H` x)S(H` D))
29	26, 28	syl5bir 227	. . . . . . . . . . . 12 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D})))
30	29	exp32 408	. . . . . . . . . . 11 |- (H Isom R, S (A, B) -> (x e. A -> (D e. A -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D})))))
31	1, 30	syl9r 72	. . . . . . . . . 10 |- (H Isom R, S (A, B) -> (C C_ A -> (x e. C -> (D e. A -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D}))))))
32	31	com34 40	. . . . . . . . 9 |- (H Isom R, S (A, B) -> (C C_ A -> (D e. A -> (x e. C -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D}))))))
33	32	imp32 390	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (x e. C -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D}))))
34	33	reximdvai 2201	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (E.x e. C ((H` x) = y /\ yS(H` D)) -> E.x e. C x e. (`'R"{D})))
35	21, 34	sylbid 220	. . . . . 6 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (y e. ((H"C) i^i (`'S"{(H` D)})) -> E.x e. C x e. (`'R"{D})))
36		elin 2786	. . . . . . . 8 |- (x e. (C i^i (`'R"{D})) <-> (x e. C /\ x e. (`'R"{D})))
37	36	exbii 1398	. . . . . . 7 |- (E.x x e. (C i^i (`'R"{D})) <-> E.x(x e. C /\ x e. (`'R"{D})))
38		neq0 2885	. . . . . . 7 |- (-. (C i^i (`'R"{D})) = (/) <-> E.x x e. (C i^i (`'R"{D})))
39		df-rex 2110	. . . . . . 7 |- (E.x e. C x e. (`'R"{D}) <-> E.x(x e. C /\ x e. (`'R"{D})))
40	37, 38, 39	3bitr4i 200	. . . . . 6 |- (-. (C i^i (`'R"{D})) = (/) <-> E.x e. C x e. (`'R"{D}))
41	35, 40	syl6ibr 230	. . . . 5 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (y e. ((H"C) i^i (`'S"{(H` D)})) -> -. (C i^i (`'R"{D})) = (/)))
42	41	19.23adv 1584	. . . 4 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (E.y y e. ((H"C) i^i (`'S"{(H` D)})) -> -. (C i^i (`'R"{D})) = (/)))
43		neq0 2885	. . . 4 |- (-. ((H"C) i^i (`'S"{(H` D)})) = (/) <-> E.y y e. ((H"C) i^i (`'S"{(H` D)})))
44	42, 43	syl5ib 223	. . 3 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (-. ((H"C) i^i (`'S"{(H` D)})) = (/) -> -. (C i^i (`'R"{D})) = (/)))
45	44	con4d 91	. 2 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> ((C i^i (`'R"{D})) = (/) -> ((H"C) i^i (`'S"{(H` D)})) = (/)))
46	1	com12 14	. . . . . . . . . . . . 13 |- (x e. C -> (C C_ A -> x e. A))
47		funfvima 4828	. . . . . . . . . . . . . . . 16 |- ((Fun H /\ x e. dom H) -> (x e. C -> (H` x) e. (H"C)))
48	47	funfni 4513	. . . . . . . . . . . . . . 15 |- ((H Fn A /\ x e. A) -> (x e. C -> (H` x) e. (H"C)))
49	48	ex 402	. . . . . . . . . . . . . 14 |- (H Fn A -> (x e. A -> (x e. C -> (H` x) e. (H"C))))
50	49	com13 37	. . . . . . . . . . . . 13 |- (x e. C -> (x e. A -> (H Fn A -> (H` x) e. (H"C))))
51	46, 50	syld 30	. . . . . . . . . . . 12 |- (x e. C -> (C C_ A -> (H Fn A -> (H` x) e. (H"C))))
52	51	com13 37	. . . . . . . . . . 11 |- (H Fn A -> (C C_ A -> (x e. C -> (H` x) e. (H"C))))
53	52	imp 377	. . . . . . . . . 10 |- ((H Fn A /\ C C_ A) -> (x e. C -> (H` x) e. (H"C)))
54	53	adantrr 431	. . . . . . . . 9 |- ((H Fn A /\ (C C_ A /\ D e. A)) -> (x e. C -> (H` x) e. (H"C)))
55	2, 3	syl 12	. . . . . . . . 9 |- (H Isom R, S (A, B) -> H Fn A)
56	54, 55	sylan 497	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (x e. C -> (H` x) e. (H"C)))
57	56	adantrd 427	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> ((x e. C /\ x e. (`'R"{D})) -> (H` x) e. (H"C)))
58	25	biimpd 170	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (xRD -> (H` x)S(H` D)))
59		fvex 4689	. . . . . . . . . . . . . . . 16 |- (H` x) e. _V
60	59	eliniseg 4294	. . . . . . . . . . . . . . 15 |- ((H` D) e. _V -> ((H` x) e. (`'S"{(H` D)}) <-> (H` x)S(H` D)))
61	13, 60	ax-mp 7	. . . . . . . . . . . . . 14 |- ((H` x) e. (`'S"{(H` D)}) <-> (H` x)S(H` D))
62	58, 61	syl6ibr 230	. . . . . . . . . . . . 13 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (xRD -> (H` x) e. (`'S"{(H` D)})))
63	24, 62	sylbid 220	. . . . . . . . . . . 12 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (x e. (`'R"{D}) -> (H` x) e. (`'S"{(H` D)})))
64	63	exp32 408	. . . . . . . . . . 11 |- (H Isom R, S (A, B) -> (x e. A -> (D e. A -> (x e. (`'R"{D}) -> (H` x) e. (`'S"{(H` D)})))))
65	1, 64	syl9r 72	. . . . . . . . . 10 |- (H Isom R, S (A, B) -> (C C_ A -> (x e. C -> (D e. A -> (x e. (`'R"{D}) -> (H` x) e. (`'S"{(H` D)}))))))
66	65	com34 40	. . . . . . . . 9 |- (H Isom R, S (A, B) -> (C C_ A -> (D e. A -> (x e. C -> (x e. (`'R"{D}) -> (H` x) e. (`'S"{(H` D)}))))))
67	66	imp32 390	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (x e. C -> (x e. (`'R"{D}) -> (H` x) e. (`'S"{(H` D)}))))
68	67	imp3a 388	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> ((x e. C /\ x e. (`'R"{D})) -> (H` x) e. (`'S"{(H` D)})))
69	57, 68	jcad 661	. . . . . 6 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> ((x e. C /\ x e. (`'R"{D})) -> ((H` x) e. (H"C) /\ (H` x) e. (`'S"{(H` D)}))))
70		elin 2786	. . . . . 6 |- ((H` x) e. ((H"C) i^i (`'S"{(H` D)})) <-> ((H` x) e. (H"C) /\ (H` x) e. (`'S"{(H` D)})))
71	69, 36, 70	3imtr4g 612	. . . . 5 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (x e. (C i^i (`'R"{D})) -> (H` x) e. ((H"C) i^i (`'S"{(H` D)}))))
72		n0i 2880	. . . . 5 |- ((H` x) e. ((H"C) i^i (`'S"{(H` D)})) -> -. ((H"C) i^i (`'S"{(H` D)})) = (/))
73	71, 72	syl6 25	. . . 4 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (x e. (C i^i (`'R"{D})) -> -. ((H"C) i^i (`'S"{(H` D)})) = (/)))
74	73	19.23adv 1584	. . 3 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (E.x x e. (C i^i (`'R"{D})) -> -. ((H"C) i^i (`'S"{(H` D)})) = (/)))
75	74, 38	syl5ib 223	. 2 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (-. (C i^i (`'R"{D})) = (/) -> -. ((H"C) i^i (`'S"{(H` D)})) = (/)))
76	45, 75	impcon4bid 578	1 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> ((C i^i (`'R"{D})) = (/) <-> ((H"C) i^i (`'S"{(H` D)})) = (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338  `'ccnv 3985  "cima 3989   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998   Isom wiso 3999

This theorem is referenced by:  isofrlem 4878

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-f1o 4013  df-fv 4014  df-iso 4015

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