Theorem isoini 5378

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 3099	. . . 4 ⊢ (y ∈ (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (y ∈ B ∧ y ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
2		isof1o 5368	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A–1-1-onto→B)
3		f1ofo 5054	. . . . . . . . 9 ⊢ (𝐻:A–1-1-onto→B → 𝐻:A–onto→B)
4		forn 5030	. . . . . . . . . 10 ⊢ (𝐻:A–onto→B → ran 𝐻 = B)
5	4	eleq2d 2085	. . . . . . . . 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 5048	. . . . . . . . 9 ⊢ (𝐻:A–1-1-onto→B → 𝐻 Fn A)
8		fvelrnb 5142	. . . . . . . . 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 5113	. . . . . . . 8 ⊢ ((Fun 𝐻 ∧ 𝐷 ∈ dom 𝐻) → (𝐻‘𝐷) ∈ V)
15	14	funfni 4921	. . . . . . 7 ⊢ ((𝐻 Fn A ∧ 𝐷 ∈ A) → (𝐻‘𝐷) ∈ V)
16		vex 2534	. . . . . . . 8 ⊢ y ∈ V
17	16	eliniseg 4618	. . . . . . 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 445	. . . . 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 3099	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∩ (◡𝑅 “ {𝐷})) ↔ (x ∈ A ∧ x ∈ (◡𝑅 “ {𝐷})))
21		vex 2534	. . . . . . . . . . . . . 14 ⊢ x ∈ V
22	21	eliniseg 4618	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ A → (x ∈ (◡𝑅 “ {𝐷}) ↔ x𝑅𝐷))
23	22	anbi2d 440	. . . . . . . . . . . 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 441	. . . . . . . . . 10 ⊢ (𝐷 ∈ A → ((x ∈ (A ∩ (◡𝑅 “ {𝐷})) ∧ x𝐻y) ↔ ((x ∈ A ∧ x𝑅𝐷) ∧ x𝐻y)))
26		anass 383	. . . . . . . . . 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 5369	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (x ∈ A ∧ 𝐷 ∈ A)) → (x𝑅𝐷 ↔ (𝐻‘x)𝑆(𝐻‘𝐷)))
30		fnbrfvb 5135	. . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (x ∈ A ∧ 𝐷 ∈ A)) → (x𝐻y ↔ (𝐻‘x) = y))
34	29, 33	anbi12d 445	. . . . . . . . . . . . 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 3737	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘x) = y → ((𝐻‘x)𝑆(𝐻‘𝐷) ↔ y𝑆(𝐻‘𝐷)))
37	36	pm5.32i 430	. . . . . . . . . . . . . 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 426	. . . . . . . 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 2303	. . . . . 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 2440	. . . . . 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 2134	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = {y ∣ ∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y})
51		dfima2 4593	. 2 ⊢ (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = {y ∣ ∃x ∈ (A ∩ (◡𝑅 “ {𝐷}))x𝐻y}
52	50, 51	syl6reqr 2069	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 1226   ∈ wcel 1370  {cab 2004  ∃wrex 2281  Vcvv 2531   ∩ cin 2889  {csn 3346   class class class wbr 3734  ^◡ccnv 4267  ran crn 4269   “ cima 4271   Fn wfn 4820  –onto→wfo 4823  –1-1-onto→wf1o 4824  ‘cfv 4825   Isom wiso 4826

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-isom 4834

This theorem is referenced by:  isoini2  5379  isoselem  5380