Theorem isomin 5497

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. (Contributed by set.mm contributors, 19-Apr-2004.)

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

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel2 3269	. . . . . . . 8 |- ((C C_ A /\ y e. C) -> y e. A)
2	1	anim1i 551	. . . . . . 7 |- (((C C_ A /\ y e. C) /\ D e. A) -> (y e. A /\ D e. A))
3	2	an32s 779	. . . . . 6 |- (((C C_ A /\ D e. A) /\ y e. C) -> (y e. A /\ D e. A))
4		isorel 5490	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> (yRD <-> (H` y)S(H` D)))
5		fvex 5340	. . . . . . . . 9 |- (H` y) e. _V
6		breq1 4643	. . . . . . . . 9 |- (x = (H` y) -> (xS(H` D) <-> (H` y)S(H` D)))
7	5, 6	ceqsexv 2895	. . . . . . . 8 |- (E.x(x = (H` y) /\ xS(H` D)) <-> (H` y)S(H` D))
8		eqcom 2355	. . . . . . . . . . 11 |- (x = (H` y) <-> (H` y) = x)
9		isof1o 5489	. . . . . . . . . . . . 13 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
10		f1ofn 5289	. . . . . . . . . . . . 13 |- (H:A-1-1-onto->B -> H Fn A)
11	9, 10	syl 15	. . . . . . . . . . . 12 |- (H Isom R, S (A, B) -> H Fn A)
12		simpl 443	. . . . . . . . . . . 12 |- ((y e. A /\ D e. A) -> y e. A)
13		fnbrfvb 5359	. . . . . . . . . . . 12 |- ((H Fn A /\ y e. A) -> ((H` y) = x <-> yHx))
14	11, 12, 13	syl2an 463	. . . . . . . . . . 11 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> ((H` y) = x <-> yHx))
15	8, 14	syl5bb 248	. . . . . . . . . 10 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> (x = (H` y) <-> yHx))
16	15	anbi1d 685	. . . . . . . . 9 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> ((x = (H` y) /\ xS(H` D)) <-> (yHx /\ xS(H` D))))
17	16	exbidv 1626	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> (E.x(x = (H` y) /\ xS(H` D)) <-> E.x(yHx /\ xS(H` D))))
18	7, 17	syl5bbr 250	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> ((H` y)S(H` D) <-> E.x(yHx /\ xS(H` D))))
19	4, 18	bitrd 244	. . . . . 6 |- ((H Isom R, S (A, B) /\ (y e. A /\ D e. A)) -> (yRD <-> E.x(yHx /\ xS(H` D))))
20	3, 19	sylan2 460	. . . . 5 |- ((H Isom R, S (A, B) /\ ((C C_ A /\ D e. A) /\ y e. C)) -> (yRD <-> E.x(yHx /\ xS(H` D))))
21	20	anassrs 629	. . . 4 |- (((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) /\ y e. C) -> (yRD <-> E.x(yHx /\ xS(H` D))))
22	21	rexbidva 2632	. . 3 |- ((H Isom R, S (A, B) /\ (C C_ A /\ D e. A)) -> (E.y e. C yRD <-> E.y e. C E.x(yHx /\ xS(H` D))))
23		elin 3220	. . . . . 6 |- (y e. (C i^i (`'R"{D})) <-> (y e. C /\ y e. (`'R"{D})))
24		eliniseg 5021	. . . . . . 7 |- (y e. (`'R"{D}) <-> yRD)
25	24	anbi2i 675	. . . . . 6 |- ((y e. C /\ y e. (`'R"{D})) <-> (y e. C /\ yRD))
26	23, 25	bitri 240	. . . . 5 |- (y e. (C i^i (`'R"{D})) <-> (y e. C /\ yRD))
27	26	exbii 1582	. . . 4 |- (E.y y e. (C i^i (`'R"{D})) <-> E.y(y e. C /\ yRD))
28		neq0 3561	. . . 4 |- (-. (C i^i (`'R"{D})) = (/) <-> E.y y e. (C i^i (`'R"{D})))
29		df-rex 2621	. . . 4 |- (E.y e. C yRD <-> E.y(y e. C /\ yRD))
30	27, 28, 29	3bitr4i 268	. . 3 |- (-. (C i^i (`'R"{D})) = (/) <-> E.y e. C yRD)
31		elima 4755	. . . . . . 7 |- (x e. (H"C) <-> E.y e. C yHx)
32		eliniseg 5021	. . . . . . 7 |- (x e. (`'S"{(H` D)}) <-> xS(H` D))
33	31, 32	anbi12i 678	. . . . . 6 |- ((x e. (H"C) /\ x e. (`'S"{(H` D)})) <-> (E.y e. C yHx /\ xS(H` D)))
34		elin 3220	. . . . . 6 |- (x e. ((H"C) i^i (`'S"{(H` D)})) <-> (x e. (H"C) /\ x e. (`'S"{(H` D)})))
35		r19.41v 2765	. . . . . 6 |- (E.y e. C (yHx /\ xS(H` D)) <-> (E.y e. C yHx /\ xS(H` D)))
36	33, 34, 35	3bitr4i 268	. . . . 5 |- (x e. ((H"C) i^i (`'S"{(H` D)})) <-> E.y e. C (yHx /\ xS(H` D)))
37	36	exbii 1582	. . . 4 |- (E.x x e. ((H"C) i^i (`'S"{(H` D)})) <-> E.xE.y e. C (yHx /\ xS(H` D)))
38		neq0 3561	. . . 4 |- (-. ((H"C) i^i (`'S"{(H` D)})) = (/) <-> E.x x e. ((H"C) i^i (`'S"{(H` D)})))
39		rexcom4 2879	. . . 4 |- (E.y e. C E.x(yHx /\ xS(H` D)) <-> E.xE.y e. C (yHx /\ xS(H` D)))
40	37, 38, 39	3bitr4i 268	. . 3 |- (-. ((H"C) i^i (`'S"{(H` D)})) = (/) <-> E.y e. C E.x(yHx /\ xS(H` D)))
41	22, 30, 40	3bitr4g 279	. 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)})) = (/)))
42	41	con4bid 284	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 3   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616   i^i cin 3209   C_ wss 3258  (/)c0 3551  {csn 3738   class class class wbr 4640  "cima 4723  `'ccnv 4772   Fn wfn 4777  -1-1-onto->wf1o 4781  ` cfv 4782   Isom wiso 4783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is referenced by: (None)