Theorem isomin 3905

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 (_ 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 2066	. . . . . . . . . . . . . 14 |- (C (_ A -> (x e. C -> x e. A))
2		isof1o 3899	. . . . . . . . . . . . . . 15 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
3		f1ofn 3696	. . . . . . . . . . . . . . 15 |- (H:A-1-1-onto->B -> H Fn A)
4		visset 1816	. . . . . . . . . . . . . . . . 17 |- y e. V
5	4	fnbrfvb 3759	. . . . . . . . . . . . . . . 16 |- ((H Fn A /\ x e. A) -> ((H` x) = y <-> xHy))
6	5	ex 373	. . . . . . . . . . . . . . 15 |- (H Fn A -> (x e. A -> ((H` x) = y <-> xHy)))
7	2, 3, 6	3syl 20	. . . . . . . . . . . . . 14 |- (H Isom R, S (A, B) -> (x e. A -> ((H` x) = y <-> xHy)))
8	1, 7	syl9r 58	. . . . . . . . . . . . 13 |- (H Isom R, S (A, B) -> (C (_ A -> (x e. C -> ((H` x) = y <-> xHy))))
9	8	imp31 362	. . . . . . . . . . . 12 |- (((H Isom R, S (A, B) /\ C (_ A) /\ x e. C) -> ((H` x) = y <-> xHy))
10	9	rexbidva 1663	. . . . . . . . . . 11 |- ((H Isom R, S (A, B) /\ C (_ A) -> (E.x e. C (H` x) = y <-> E.x e. C xHy))
11	4	elima 3414	. . . . . . . . . . 11 |- (y e. (H"C) <-> E.x e. C xHy)
12	10, 11	syl6rbbr 541	. . . . . . . . . 10 |- ((H Isom R, S (A, B) /\ C (_ A) -> (y e. (H"C) <-> E.x e. C (H` x) = y))
13		fvex 3738	. . . . . . . . . . . 12 |- (H` D) e. V
14	4	eliniseg 3435	. . . . . . . . . . . 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 (_ A) -> (y e. (`'S"{(H` D)}) <-> yS(H` D)))
17	12, 16	anbi12d 630	. . . . . . . . 9 |- ((H Isom R, S (A, B) /\ C (_ A) -> ((y e. (H"C) /\ y e. (`'S"{(H` D)})) <-> (E.x e. C (H` x) = y /\ yS(H` D))))
18		elin 2210	. . . . . . . . 9 |- (y e. ((H"C) i^i (`'S"{(H` D)})) <-> (y e. (H"C) /\ y e. (`'S"{(H` D)})))
19		r19.41v 1766	. . . . . . . . 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 557	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ C (_ A) -> (y e. ((H"C) i^i (`'S"{(H` D)})) <-> E.x e. C ((H` x) = y /\ yS(H` D))))
21	20	adantrr 397	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (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 1816	. . . . . . . . . . . . . . . 16 |- x e. V
23	22	eliniseg 3435	. . . . . . . . . . . . . . 15 |- (D e. A -> (x e. (`'R"{D}) <-> xRD))
24	23	ad2antll 409	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (x e. (`'R"{D}) <-> xRD))
25		isorel 3900	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (xRD <-> (H` x)S(H` D)))
26	24, 25	bitrd 530	. . . . . . . . . . . . 13 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (x e. (`'R"{D}) <-> (H` x)S(H` D)))
27		breq1 2627	. . . . . . . . . . . . . 14 |- ((H` x) = y -> ((H` x)S(H` D) <-> yS(H` D)))
28	27	biimpar 419	. . . . . . . . . . . . 13 |- (((H` x) = y /\ yS(H` D)) -> (H` x)S(H` D))
29	26, 28	syl5bir 210	. . . . . . . . . . . 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 379	. . . . . . . . . . 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 58	. . . . . . . . . 10 |- (H Isom R, S (A, B) -> (C (_ A -> (x e. C -> (D e. A -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D}))))))
32	31	com34 36	. . . . . . . . 9 |- (H Isom R, S (A, B) -> (C (_ A -> (D e. A -> (x e. C -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D}))))))
33	32	imp32 363	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ (C (_ A /\ D e. A)) -> (x e. C -> (((H` x) = y /\ yS(H` D)) -> x e. (`'R"{D}))))
34	33	r19.22dv 1740	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (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 203	. . . . . 6 |- ((H Isom R, S (A, B) /\ (C (_ A /\ D e. A)) -> (y e. ((H"C) i^i (`'S"{(H` D)})) -> E.x e. C x e. (`'R"{D})))
36		elin 2210	. . . . . . . 8 |- (x e. (C i^i (`'R"{D})) <-> (x e. C /\ x e. (`'R"{D})))
37	36	exbii 1053	. . . . . . 7 |- (E.x x e. (C i^i (`'R"{D})) <-> E.x(x e. C /\ x e. (`'R"{D})))
38		n0 2293	. . . . . . 7 |- (-. (C i^i (`'R"{D})) = (/) <-> E.x x e. (C i^i (`'R"{D})))
39		df-rex 1653	. . . . . . 7 |- (E.x e. C x e. (`'R"{D}) <-> E.x(x e. C /\ x e. (`'R"{D})))
40	37, 38, 39	3bitr4 183	. . . . . 6 |- (-. (C i^i (`'R"{D})) = (/) <-> E.x e. C x e. (`'R"{D}))
41	35, 40	syl6ibr 213	. . . . 5 |- ((H Isom R, S (A, B) /\ (C (_ A /\ D e. A)) -> (y e. ((H"C) i^i (`'S"{(H` D)})) -> -. (C i^i (`'R"{D})) = (/)))
42	41	19.23adv 1216	. . . 4 |- ((H Isom R, S (A, B) /\ (C (_ A /\ D e. A)) -> (E.y y e. ((H"C) i^i (`'S"{(H` D)})) -> -. (C i^i (`'R"{D})) = (/)))
43		n0 2293	. . . 4 |- (-. ((H"C) i^i (`'S"{(H` D)})) = (/) <-> E.y y e. ((H"C) i^i (`'S"{(H` D)})))
44	42, 43	syl5ib 206	. . 3 |- ((H Isom R, S (A, B) /\ (C (_ A /\ D e. A)) -> (-. ((H"C) i^i (`'S"{(H` D)})) = (/) -> -. (C i^i (`'R"{D})) = (/)))
45	44	a3d 75	. 2 |- ((H Isom R, S (A, B) /\ (C (_ A /\ D e. A)) -> ((C i^i (`'R"{D})) = (/) -> ((H"C) i^i (`'S"{(H` D)})) = (/)))
46	1	com12 11	. . . . . . . . . . . . . 14 |- (x e. C -> (C (_ A -> x e. A))
47		funfvima 3858	. . . . . . . . . . . . . . . . 17 |- ((Fun H /\ x e. dom H) -> (x e. C -> (H` x) e. (H"C)))
48	47	funfni 3594	. . . . . . . . . . . . . . . 16 |- ((H Fn A /\ x e. A) -> (x e. C -> (H` x) e. (H"C)))
49	48	ex 373	. . . . . . . . . . . . . . 15 |- (H Fn