Theorem isoini 7358

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 6080	. 2 ⊢ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦}
2		elin 3967	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
3		isof1o 7343	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1ofo 6855	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
5		forn 6823	. . . . . . . . . 10 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
6	5	eleq2d 2827	. . . . . . . . 9 ⊢ (𝐻:𝐴–onto→𝐵 → (𝑦 ∈ ran 𝐻 ↔ 𝑦 ∈ 𝐵))
7	3, 4, 6	3syl 18	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ 𝑦 ∈ 𝐵))
8		f1ofn 6849	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
9		fvelrnb 6969	. . . . . . . . 9 ⊢ (𝐻 Fn 𝐴 → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
10	3, 8, 9	3syl 18	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
11	7, 10	bitr3d 281	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
12		fvex 6919	. . . . . . . 8 ⊢ (𝐻‘𝐷) ∈ V
13		vex 3484	. . . . . . . . 9 ⊢ 𝑦 ∈ V
14	13	eliniseg 6112	. . . . . . . 8 ⊢ ((𝐻‘𝐷) ∈ V → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
15	12, 14	mp1i 13	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
16	11, 15	anbi12d 632	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
17	16	adantr 480	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
18		elin 3967	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
19		vex 3484	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
20	19	eliniseg 6112	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
21	20	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
22	18, 21	bitrid 283	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
23	22	anbi1d 631	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
24		anass 468	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦)))
25	23, 24	bitrdi 287	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
26	25	adantl 481	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
27		isorel 7346	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
28	3, 8	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
29		fnbrfvb 6959	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
30	29	bicomd 223	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
31	28, 30	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
32	31	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
33	27, 32	anbi12d 632	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦)))
34		ancom 460	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
35		breq1 5146	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑥) = 𝑦 → ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ↔ 𝑦𝑆(𝐻‘𝐷)))
36	35	pm5.32i 574	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
37	34, 36	bitri 275	. . . . . . . . . . . . 13 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
38	33, 37	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
39	38	exp32 420	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))))
40	39	com23 86	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))))
41	40	imp 406	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
42	41	pm5.32d 577	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
43	26, 42	bitrd 279	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
44	43	rexbidv2 3175	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥 ∈ 𝐴 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
45		r19.41v 3189	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
46	44, 45	bitrdi 287	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
47	17, 46	bitr4d 282	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
48	2, 47	bitrid 283	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
49	48	eqabdv 2875	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦})
50	1, 49	eqtr4id 2796	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480   ∩ cin 3950  {csn 4626   class class class wbr 5143  ^◡ccnv 5684  ran crn 5686   “ cima 5688   Fn wfn 6556  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561   Isom wiso 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570

This theorem is referenced by:  isoini2  7359  isoselem  7361  infxpenlem  10053