Theorem isoini 5498

Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 20-Apr-2004.)

Assertion

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

Proof of Theorem isoini

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

Step	Hyp	Ref	Expression
1		elin 3220	. . . . 5 |- (y e. (B i^i (`'S"{(H` D)})) <-> (y e. B /\ y e. (`'S"{(H` D)})))
2		eliniseg 5021	. . . . . 6 |- (y e. (`'S"{(H` D)}) <-> yS(H` D))
3	2	anbi2i 675	. . . . 5 |- ((y e. B /\ y e. (`'S"{(H` D)})) <-> (y e. B /\ yS(H` D)))
4	1, 3	bitri 240	. . . 4 |- (y e. (B i^i (`'S"{(H` D)})) <-> (y e. B /\ yS(H` D)))
5		isof1o 5489	. . . . . . . . . . 11 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
6		f1ofo 5294	. . . . . . . . . . 11 |- (H:A-1-1-onto->B -> H:A-onto->B)
7	5, 6	syl 15	. . . . . . . . . 10 |- (H Isom R, S (A, B) -> H:A-onto->B)
8		forn 5273	. . . . . . . . . 10 |- (H:A-onto->B -> ran H = B)
9	7, 8	syl 15	. . . . . . . . 9 |- (H Isom R, S (A, B) -> ran H = B)
10	9	eleq2d 2420	. . . . . . . 8 |- (H Isom R, S (A, B) -> (y e. ran H <-> y e. B))
11		f1ofn 5289	. . . . . . . . . 10 |- (H:A-1-1-onto->B -> H Fn A)
12	5, 11	syl 15	. . . . . . . . 9 |- (H Isom R, S (A, B) -> H Fn A)
13		fvelrnb 5366	. . . . . . . . 9 |- (H Fn A -> (y e. ran H <-> E.x e. A (H` x) = y))
14	12, 13	syl 15	. . . . . . . 8 |- (H Isom R, S (A, B) -> (y e. ran H <-> E.x e. A (H` x) = y))
15	10, 14	bitr3d 246	. . . . . . 7 |- (H Isom R, S (A, B) -> (y e. B <-> E.x e. A (H` x) = y))
16	15	anbi1d 685	. . . . . 6 |- (H Isom R, S (A, B) -> ((y e. B /\ yS(H` D)) <-> (E.x e. A (H` x) = y /\ yS(H` D))))
17	16	adantr 451	. . . . 5 |- ((H Isom R, S (A, B) /\ D e. A) -> ((y e. B /\ yS(H` D)) <-> (E.x e. A (H` x) = y /\ yS(H` D))))
18		elin 3220	. . . . . . . . . . 11 |- (x e. (A i^i (`'R"{D})) <-> (x e. A /\ x e. (`'R"{D})))
19		eliniseg 5021	. . . . . . . . . . . 12 |- (x e. (`'R"{D}) <-> xRD)
20	19	anbi2i 675	. . . . . . . . . . 11 |- ((x e. A /\ x e. (`'R"{D})) <-> (x e. A /\ xRD))
21	18, 20	bitri 240	. . . . . . . . . 10 |- (x e. (A i^i (`'R"{D})) <-> (x e. A /\ xRD))
22	21	anbi1i 676	. . . . . . . . 9 |- ((x e. (A i^i (`'R"{D})) /\ xHy) <-> ((x e. A /\ xRD) /\ xHy))
23		anass 630	. . . . . . . . 9 |- (((x e. A /\ xRD) /\ xHy) <-> (x e. A /\ (xRD /\ xHy)))
24	22, 23	bitri 240	. . . . . . . 8 |- ((x e. (A i^i (`'R"{D})) /\ xHy) <-> (x e. A /\ (xRD /\ xHy)))
25		fnbrfvb 5359	. . . . . . . . . . . . . . . . 17 |- ((H Fn A /\ x e. A) -> ((H` x) = y <-> xHy))
26	12, 25	sylan 457	. . . . . . . . . . . . . . . 16 |- ((H Isom R, S (A, B) /\ x e. A) -> ((H` x) = y <-> xHy))
27	26	adantrr 697	. . . . . . . . . . . . . . 15 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> ((H` x) = y <-> xHy))
28	27	bicomd 192	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (xHy <-> (H` x) = y))
29		isorel 5490	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> (xRD <-> (H` x)S(H` D)))
30	28, 29	anbi12d 691	. . . . . . . . . . . . 13 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> ((xHy /\ xRD) <-> ((H` x) = y /\ (H` x)S(H` D))))
31		ancom 437	. . . . . . . . . . . . 13 |- ((xHy /\ xRD) <-> (xRD /\ xHy))
32		breq1 4643	. . . . . . . . . . . . . 14 |- ((H` x) = y -> ((H` x)S(H` D) <-> yS(H` D)))
33	32	pm5.32i 618	. . . . . . . . . . . . 13 |- (((H` x) = y /\ (H` x)S(H` D)) <-> ((H` x) = y /\ yS(H` D)))
34	30, 31, 33	3bitr3g 278	. . . . . . . . . . . 12 |- ((H Isom R, S (A, B) /\ (x e. A /\ D e. A)) -> ((xRD /\ xHy) <-> ((H` x) = y /\ yS(H` D))))
35	34	exp32 588	. . . . . . . . . . 11 |- (H Isom R, S (A, B) -> (x e. A -> (D e. A -> ((xRD /\ xHy) <-> ((H` x) = y /\ yS(H` D))))))
36	35	com23 72	. . . . . . . . . 10 |- (H Isom R, S (A, B) -> (D e. A -> (x e. A -> ((xRD /\ xHy) <-> ((H` x) = y /\ yS(H` D))))))
37	36	imp 418	. . . . . . . . 9 |- ((H Isom R, S (A, B) /\ D e. A) -> (x e. A -> ((xRD /\ xHy) <-> ((H` x) = y /\ yS(H` D)))))
38	37	pm5.32d 620	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ D e. A) -> ((x e. A /\ (xRD /\ xHy)) <-> (x e. A /\ ((H` x) = y /\ yS(H` D)))))
39	24, 38	syl5bb 248	. . . . . . 7 |- ((H Isom R, S (A, B) /\ D e. A) -> ((x e. (A i^i (`'R"{D})) /\ xHy) <-> (x e. A /\ ((H` x) = y /\ yS(H` D)))))
40	39	rexbidv2 2638	. . . . . 6 |- ((H Isom R, S (A, B) /\ D e. A) -> (E.x e. (A i^i (`'R"{D}))xHy <-> E.x e. A ((H` x) = y /\ yS(H` D))))
41		r19.41v 2765	. . . . . 6 |- (E.x e. A ((H` x) = y /\ yS(H` D)) <-> (E.x e. A (H` x) = y /\ yS(H` D)))
42	40, 41	syl6bb 252	. . . . 5 |- ((H Isom R, S (A, B) /\ D e. A) -> (E.x e. (A i^i (`'R"{D}))xHy <-> (E.x e. A (H` x) = y /\ yS(H` D))))
43	17, 42	bitr4d 247	. . . 4 |- ((H Isom R, S (A, B) /\ D e. A) -> ((y e. B /\ yS(H` D)) <-> E.x e. (A i^i (`'R"{D}))xHy))
44	4, 43	syl5bb 248	. . 3 |- ((H Isom R, S (A, B) /\ D e. A) -> (y e. (B i^i (`'S"{(H` D)})) <-> E.x e. (A i^i (`'R"{D}))xHy))
45	44	abbi2dv 2469	. 2 |- ((H Isom R, S (A, B) /\ D e. A) -> (B i^i (`'S"{(H` D)})) = {y | E.x e. (A i^i (`'R"{D}))xHy})
46		df-ima 4728	. 2 |- (H"(A i^i (`'R"{D}))) = {y | E.x e. (A i^i (`'R"{D}))xHy}
47	45, 46	syl6reqr 2404	1 |- ((H Isom R, S (A, B) /\ D e. A) -> (H"(A i^i (`'R"{D}))) = (B i^i (`'S"{(H` D)})))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616   i^i cin 3209  {csn 3738   class class class wbr 4640  "cima 4723  `'ccnv 4772  ran crn 4774   Fn wfn 4777  -onto->wfo 4780  -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-fo 4794  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is referenced by:  isoini2  5499