Theorem isoini 5400

Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 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 3120	. . . 4 |- (y e. (B i^i ( `' S " {( H ` D) })) <-> (y e. B /\ y e. ( `' S " {( H ` D) })))
2		isof1o 5390	. . . . . . . . 9 |- ( H Isom R , S (A , B) -> H :A -1-1-onto->B)
3		f1ofo 5076	. . . . . . . . 9 |- ( H :A -1-1-onto->B -> H :A -onto->B)
4		forn 5052	. . . . . . . . . 10 |- ( H :A -onto->B -> ran H = B)
5	4	eleq2d 2104	. . . . . . . . 9 |- ( H :A -onto->B -> (y e. ran H <-> y e. B))
6	2, 3, 5	3syl 17	. . . . . . . 8 |- ( H Isom R , S (A , B) -> (y e. ran H <-> y e. B))
7		f1ofn 5070	. . . . . . . . 9 |- ( H :A -1-1-onto->B -> H Fn A)
8		fvelrnb 5164	. . . . . . . . 9 |- ( H Fn A -> (y e. ran H <-> E.x e. A ( H ` x) = y))
9	2, 7, 8	3syl 17	. . . . . . . 8 |- ( H Isom R , S (A , B) -> (y e. ran H <-> E.x e. A ( H ` x) = y))
10	6, 9	bitr3d 179	. . . . . . 7 |- ( H Isom R , S (A , B) -> (y e. B <-> E.x e. A ( H ` x) = y))
11	10	adantr 261	. . . . . 6 |- (( H Isom R , S (A , B) /\ D e. A) -> (y e. B <-> E.x e. A ( H ` x) = y))
12	2, 7	syl 14	. . . . . . . 8 |- ( H Isom R , S (A , B) -> H Fn A)
13	12	anim1i 323	. . . . . . 7 |- (( H Isom R , S (A , B) /\ D e. A) -> ( H Fn A /\ D e. A))
14		funfvex 5135	. . . . . . . 8 |- (( Fun H /\ D e. dom H) -> ( H ` D) e. _V)
15	14	funfni 4942	. . . . . . 7 |- (( H Fn A /\ D e. A) -> ( H ` D) e. _V)
16		vex 2554	. . . . . . . 8 |- y e. _V
17	16	eliniseg 4638	. . . . . . 7 |- (( H ` D) e. _V -> (y e. ( `' S " {( H ` D) }) <-> y S( H ` D)))
18	13, 15, 17	3syl 17	. . . . . 6 |- (( H Isom R , S (A , B) /\ D e. A) -> (y e. ( `' S " {( H ` D) }) <-> y S( H ` D)))
19	11, 18	anbi12d 442	. . . . 5 |- (( H Isom R , S (A , B) /\ D e. A) -> ((y e. B /\ y e. ( `' S " {( H ` D) })) <-> (E.x e. A ( H ` x) = y /\ y S( H ` D))))
20		elin 3120	. . . . . . . . . . . 12 |- (x e. (A i^i ( `' R " { D })) <-> (x e. A /\ x e. ( `' R " { D })))
21		vex 2554	. . . . . . . . . . . . . 14 |- x e. _V
22	21	eliniseg 4638	. . . . . . . . . . . . 13 |- ( D e. A -> (x e. ( `' R " { D }) <-> x R D))
23	22	anbi2d 437	. . . . . . . . . . . 12 |- ( D e. A -> ((x e. A /\ x e. ( `' R " { D })) <-> (x e. A /\ x R D)))
24	20, 23	syl5bb 181	. . . . . . . . . . 11 |- ( D e. A -> (x e. (A i^i ( `' R " { D })) <-> (x e. A /\ x R D)))
25	24	anbi1d 438	. . . . . . . . . 10 |- ( D e. A -> ((x e. (A i^i ( `' R " { D })) /\ x Hy) <-> ((x e. A /\ x R D) /\ x Hy)))
26		anass 381	. . . . . . . . . 10 |- (((x e. A /\ x R D) /\ x Hy) <-> (x e. A /\ (x R D /\ x Hy)))
27	25, 26	syl6bb 185	. . . . . . . . 9 |- ( D e. A -> ((x e. (A i^i ( `' R " { D })) /\ x Hy) <-> (x e. A /\ (x R D /\ x Hy))))
28	27	adantl 262	. . . . . . . 8 |- (( H Isom R , S (A , B) /\ D e. A) -> ((x e. (A i^i ( `' R " { D })) /\ x Hy) <-> (x e. A /\ (x R D /\ x Hy))))
29		isorel 5391	. . . . . . . . . . . . . 14 |- (( H Isom R , S (A , B) /\ (x e. A /\ D e. A)) -> (x R D <-> ( H ` x) S( H ` D)))
30		fnbrfvb 5157	. . . . . . . . . . . . . . . . 17 |- (( H Fn A /\ x e. A) -> (( H ` x) = y <-> x Hy))
31	30	bicomd 129	. . . . . . . . . . . . . . . 16 |- (( H Fn A /\ x e. A) -> (x Hy <-> ( H ` x) = y))
32	12, 31	sylan 267	. . . . . . . . . . . . . . 15 |- (( H Isom R , S (A , B) /\ x e. A) -> (x Hy <-> ( H ` x) = y))
33	32	adantrr 448	. . . . . . . . . . . . . 14 |- (( H Isom R , S (A , B) /\ (x e. A /\ D e. A)) -> (x Hy <-> ( H ` x) = y))
34	29, 33	anbi12d 442	. . . . . . . . . . . . 13 |- (( H Isom R , S (A , B) /\ (x e. A /\ D e. A)) -> ((x R D /\ x Hy) <-> (( H ` x) S( H ` D) /\ ( H ` x) = y)))
35		ancom 253	. . . . . . . . . . . . . 14 |- ((( H ` x) S( H ` D) /\ ( H ` x) = y) <-> (( H ` x) = y /\ ( H ` x) S( H ` D)))
36		breq1 3758	. . . . . . . . . . . . . . 15 |- (( H ` x) = y -> (( H ` x) S( H ` D) <-> y S( H ` D)))
37	36	pm5.32i 427	. . . . . . . . . . . . . 14 |- ((( H ` x) = y /\ ( H ` x) S( H ` D)) <-> (( H ` x) = y /\ y S( H ` D)))
38	35, 37	bitri 173	. . . . . . . . . . . . 13 |- ((( H ` x) S( H ` D) /\ ( H ` x) = y) <-> (( H ` x) = y /\ y S( H ` D)))
39	34, 38	syl6bb 185	. . . . . . . . . . . 12 |- (( H Isom R , S (A , B) /\ (x e. A /\ D e. A)) -> ((x R D /\ x Hy) <-> (( H ` x) = y /\ y S( H ` D))))
40	39	exp32 347	. . . . . . . . . . 11 |- ( H Isom R , S (A , B) -> (x e. A -> ( D e. A -> ((x R D /\ x Hy) <-> (( H ` x) = y /\ y S( H ` D))))))
41	40	com23 72	. . . . . . . . . 10 |- ( H Isom R , S (A , B) -> ( D e. A -> (x e. A -> ((x R D /\ x Hy) <-> (( H ` x) = y /\ y S( H ` D))))))
42	41	imp 115	. . . . . . . . 9 |- (( H Isom R , S (A , B) /\ D e. A) -> (x e. A -> ((x R D /\ x Hy) <-> (( H ` x) = y /\ y S( H ` D)))))
43	42	pm5.32d 423	. . . . . . . 8 |- (( H Isom R , S (A , B) /\ D e. A) -> ((x e. A /\ (x R D /\ x Hy)) <-> (x e. A /\ (( H ` x) = y /\ y S( H ` D)))))
44	28, 43	bitrd 177	. . . . . . 7 |- (( H Isom R , S (A , B) /\ D e. A) -> ((x e. (A i^i ( `' R " { D })) /\ x Hy) <-> (x e. A /\ (( H ` x) = y /\ y S( H ` D)))))
45	44	rexbidv2 2323	. . . . . 6 |- (( H Isom R , S (A , B) /\ D e. A) -> (E.x e. (A i^i ( `' R " { D }))x Hy <-> E.x e. A (( H ` x) = y /\ y S( H ` D))))
46		r19.41v 2460	. . . . . 6 |- (E.x e. A (( H ` x) = y /\ y S( H ` D)) <-> (E.x e. A ( H ` x) = y /\ y S( H ` D)))
47	45, 46	syl6bb 185	. . . . 5 |- (( H Isom R , S (A , B) /\ D e. A) -> (E.x e. (A i^i ( `' R " { D }))x Hy <-> (E.x e. A ( H ` x) = y /\ y S( H ` D))))
48	19, 47	bitr4d 180	. . . 4 |- (( H Isom R , S (A , B) /\ D e. A) -> ((y e. B /\ y e. ( `' S " {( H ` D) })) <-> E.x e. (A i^i ( `' R " { D }))x Hy))
49	1, 48	syl5bb 181	. . 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 }))x Hy))
50	49	abbi2dv 2153	. 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 }))x Hy })
51		dfima2 4613	. 2 |- ( H "(A i^i ( `' R " { D }))) = {y | E.x e. (A i^i ( `' R " { D }))x Hy }
52	50, 51	syl6reqr 2088	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   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   {cab 2023  E.wrex 2301   _Vcvv 2551   i^i cin 2910   {csn 3367   class class class wbr 3755   `'ccnv 4287   ran crn 4289   "cima 4291   Fn wfn 4840   -onto->wfo 4843   -1-1-onto->wf1o 4844   ` cfv 4845   Isom wiso 4846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854

This theorem is referenced by:  isoini2  5401  isoselem  5402