Theorem isoini 5786

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		dfima2 4948	. 2 ⊢ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦}
2		elin 3305	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
3		isof1o 5775	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1ofo 5439	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
5		forn 5413	. . . . . . . . . 10 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
6	5	eleq2d 2236	. . . . . . . . 9 ⊢ (𝐻:𝐴–onto→𝐵 → (𝑦 ∈ ran 𝐻 ↔ 𝑦 ∈ 𝐵))
7	3, 4, 6	3syl 17	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ 𝑦 ∈ 𝐵))
8		f1ofn 5433	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
9		fvelrnb 5534	. . . . . . . . 9 ⊢ (𝐻 Fn 𝐴 → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
10	3, 8, 9	3syl 17	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
11	7, 10	bitr3d 189	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
12	11	adantr 274	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
13	3, 8	syl 14	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
14	13	anim1i 338	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 Fn 𝐴 ∧ 𝐷 ∈ 𝐴))
15		funfvex 5503	. . . . . . . 8 ⊢ ((Fun 𝐻 ∧ 𝐷 ∈ dom 𝐻) → (𝐻‘𝐷) ∈ V)
16	15	funfni 5288	. . . . . . 7 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐻‘𝐷) ∈ V)
17		vex 2729	. . . . . . . 8 ⊢ 𝑦 ∈ V
18	17	eliniseg 4974	. . . . . . 7 ⊢ ((𝐻‘𝐷) ∈ V → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
19	14, 16, 18	3syl 17	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
20	12, 19	anbi12d 465	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
21		elin 3305	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
22		vex 2729	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
23	22	eliniseg 4974	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
24	23	anbi2d 460	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
25	21, 24	syl5bb 191	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
26	25	anbi1d 461	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
27		anass 399	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦)))
28	26, 27	bitrdi 195	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
29	28	adantl 275	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
30		isorel 5776	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
31		fnbrfvb 5527	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
32	31	bicomd 140	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
33	13, 32	sylan 281	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
34	33	adantrr 471	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
35	30, 34	anbi12d 465	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦)))
36		ancom 264	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
37		breq1 3985	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑥) = 𝑦 → ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ↔ 𝑦𝑆(𝐻‘𝐷)))
38	37	pm5.32i 450	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
39	36, 38	bitri 183	. . . . . . . . . . . . 13 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
40	35, 39	bitrdi 195	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
41	40	exp32 363	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))))
42	41	com23 78	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))))
43	42	imp 123	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
44	43	pm5.32d 446	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
45	29, 44	bitrd 187	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
46	45	rexbidv2 2469	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥 ∈ 𝐴 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
47		r19.41v 2622	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
48	46, 47	bitrdi 195	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
49	20, 48	bitr4d 190	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
50	2, 49	syl5bb 191	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
51	50	abbi2dv 2285	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦})
52	1, 51	eqtr4id 2218	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {cab 2151  ∃wrex 2445  Vcvv 2726   ∩ cin 3115  {csn 3576   class class class wbr 3982  ^◡ccnv 4603  ran crn 4605   “ cima 4607   Fn wfn 5183  –onto→wfo 5186  –1-1-onto→wf1o 5187  ‘cfv 5188   Isom wiso 5189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197

This theorem is referenced by:  isoini2  5787  isoselem  5788