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 ∈ A) → (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = (B ∩ (◡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 ∈ (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (y ∈ B ∧ y ∈ (◡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 ∈ ran 𝐻 ↔ y ∈ B))
6	2, 3, 5	3syl 17	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (y ∈ ran 𝐻 ↔ y ∈ B))
7		f1ofn 5070	. . . . . . . . 9 ⊢ (𝐻:A–1-1-onto→B → 𝐻 Fn A)
8		fvelrnb 5164	. . . . . . . . 9 ⊢ (𝐻 Fn A → (y ∈ ran 𝐻 ↔ ∃x ∈ A (𝐻‘x) = y))
9	2, 7, 8	3syl 17	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (y ∈ ran 𝐻 ↔ ∃x ∈ A (𝐻‘x) = y))
10	6, 9	bitr3d 179	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (y ∈ B ↔ ∃x ∈ A (𝐻‘x) = y))
11	10	adantr 261	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (y ∈ B ↔ ∃x ∈ 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 ∈ A) → (𝐻 Fn A ∧ 𝐷 ∈ A))
14		funfvex 5135	. . . . . . . 8 ⊢ ((Fun 𝐻 ∧ 𝐷 ∈ dom 𝐻) → (𝐻‘𝐷) ∈ V)
15	14	funfni 4942	. . . . . . 7 ⊢ ((𝐻 Fn A ∧ 𝐷 ∈ A) → (𝐻‘𝐷) ∈ V)
16		vex 2554	. . . . . . . 8 ⊢ y ∈ V
17	16	eliniseg 4638	. . . . . . 7 ⊢ ((𝐻‘𝐷) ∈ V → (y ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ y𝑆(𝐻‘𝐷)))
18	13, 15, 17	3syl 17	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (y ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ y𝑆(𝐻‘𝐷)))
19	11, 18	anbi12d 442	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → ((y ∈ B ∧ y ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (∃x ∈ A (𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷))))
20		elin 3120	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∩ (◡𝑅 “ {𝐷})) ↔ (x ∈ A ∧ x ∈ (◡𝑅 “ {𝐷})))
21		vex 2554	. . . . . . . . . . . . . 14 ⊢ x ∈ V
22	21	eliniseg 4638	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ A → (x ∈ (◡𝑅 “ {𝐷}) ↔ x𝑅𝐷))
23	22	anbi2d 437	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ A → ((x ∈ A ∧ x ∈ (◡𝑅 “ {𝐷})) ↔ (x ∈ A ∧ x𝑅𝐷)))
24	20, 23	syl5bb 181	. . . . . . . . . . 11 ⊢ (𝐷 ∈ A → (x ∈ (A ∩ (◡𝑅 “ {𝐷})) ↔ (x ∈ A ∧ x𝑅𝐷)))
25	24	anbi1d 438	. . . . . . . . . 10 ⊢ (𝐷 ∈ A → ((x ∈ (A ∩ (◡𝑅 “ {𝐷})) ∧ x𝐻y) ↔ ((x ∈ A ∧ x𝑅𝐷) ∧ x𝐻y)))
26		anass 381	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ x𝑅𝐷) ∧ x𝐻y) ↔ (x ∈ A ∧ (x𝑅𝐷 ∧ x𝐻y)))
27	25, 26	syl6bb 185	. . . . . . . . 9 ⊢ (𝐷 ∈ A → ((x ∈ (A ∩ (◡𝑅 “ {𝐷})) ∧ x𝐻y) ↔ (x ∈ A ∧ (x𝑅𝐷 ∧ x𝐻y))))
28	27	adantl 262	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → ((x ∈ (A ∩ (◡𝑅 “ {𝐷})) ∧ x𝐻y) ↔ (x ∈ A ∧ (x𝑅𝐷 ∧ x𝐻y))))
29		isorel 5391	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (x ∈ A ∧ 𝐷 ∈ A)) → (x𝑅𝐷 ↔ (𝐻‘x)𝑆(𝐻‘𝐷)))
30		fnbrfvb 5157	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 Fn A ∧ x ∈ A) → ((𝐻‘x) = y ↔ x𝐻y))
31	30	bicomd 129	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn A ∧ x ∈ A) → (x𝐻y ↔ (𝐻‘x) = y))
32	12, 31	sylan 267	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ x ∈ A) → (x𝐻y ↔ (𝐻‘x) = y))
33	32	adantrr 448	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (x ∈ A ∧ 𝐷 ∈ A)) → (x𝐻y ↔ (𝐻‘x) = y))
34	29, 33	anbi12d 442	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (x ∈ A ∧ 𝐷 ∈ A)) → ((x𝑅𝐷 ∧ x𝐻y) ↔ ((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) ↔ y𝑆(𝐻‘𝐷)))
37	36	pm5.32i 427	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘x) = y ∧ (𝐻‘x)𝑆(𝐻‘𝐷)) ↔ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)))
38	35, 37	bitri 173	. . . . . . . . . . . . 13 ⊢ (((𝐻‘x)𝑆(𝐻‘𝐷) ∧ (𝐻‘x) = y) ↔ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)))
39	34, 38	syl6bb 185	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (x ∈ A ∧ 𝐷 ∈ A)) → ((x𝑅𝐷 ∧ x𝐻y) ↔ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷))))
40	39	exp32 347	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (x ∈ A → (𝐷 ∈ A → ((x𝑅𝐷 ∧ x𝐻y) ↔ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷))))))
41	40	com23 72	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝐷 ∈ A → (x ∈ A → ((x𝑅𝐷 ∧ x𝐻y) ↔ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷))))))
42	41	imp 115	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (x ∈ A → ((x𝑅𝐷 ∧ x𝐻y) ↔ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)))))
43	42	pm5.32d 423	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → ((x ∈ A ∧ (x𝑅𝐷 ∧ x𝐻y)) ↔ (x ∈ A ∧ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)))))
44	28, 43	bitrd 177	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → ((x ∈ (A ∩ (◡𝑅 “ {𝐷})) ∧ x𝐻y) ↔ (x ∈ A ∧ ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)))))
45	44	rexbidv2 2323	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y ↔ ∃x ∈ A ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷))))
46		r19.41v 2460	. . . . . 6 ⊢ (∃x ∈ A ((𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)) ↔ (∃x ∈ A (𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷)))
47	45, 46	syl6bb 185	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y ↔ (∃x ∈ A (𝐻‘x) = y ∧ y𝑆(𝐻‘𝐷))))
48	19, 47	bitr4d 180	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → ((y ∈ B ∧ y ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y))
49	1, 48	syl5bb 181	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (y ∈ (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y))
50	49	abbi2dv 2153	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = {y ∣ ∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y})
51		dfima2 4613	. 2 ⊢ (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = {y ∣ ∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y}
52	50, 51	syl6reqr 2088	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551   ∩ 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