Theorem isoini 7347

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 6069	. 2 ⊢ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦}
2		elin 3924	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
3		isof1o 7332	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1ofo 6835	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
5		forn 6802	. . . . . . . . . 10 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
6	5	eleq2d 2852	. . . . . . . . 9 ⊢ (𝐻:𝐴–onto→𝐵 → (𝑦 ∈ ran 𝐻 ↔ 𝑦 ∈ 𝐵))
7	3, 4, 6	3syl 19	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ 𝑦 ∈ 𝐵))
8		f1ofn 6828	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
9		fvelrnb 6948	. . . . . . . . 9 ⊢ (𝐻 Fn 𝐴 → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
10	3, 8, 9	3syl 19	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
11	7, 10	bitr3d 284	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
12		fvex 6901	. . . . . . . 8 ⊢ (𝐻‘𝐷) ∈ V
13		vex 3462	. . . . . . . . 9 ⊢ 𝑦 ∈ V
14	13	eliniseg 6101	. . . . . . . 8 ⊢ ((𝐻‘𝐷) ∈ V → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
15	12, 14	mp1i 14	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
16	11, 15	anbi12d 644	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
17	16	adantr 486	. . . . 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 3924	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
19		vex 3462	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
20	19	eliniseg 6101	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
21	20	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
22	18, 21	bitrid 286	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
23	22	anbi1d 643	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
24		anass 474	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦)))
25	23, 24	bitrdi 290	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
26	25	adantl 487	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
27		isorel 7335	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
28	3, 8	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
29		fnbrfvb 6938	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
30	29	bicomd 226	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
31	28, 30	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
32	31	adantrr 730	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
33	27, 32	anbi12d 644	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦)))
34		ancom 466	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
35		breq1 5117	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑥) = 𝑦 → ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ↔ 𝑦𝑆(𝐻‘𝐷)))
36	35	pm5.32i 585	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
37	34, 36	bitri 278	. . . . . . . . . . . . 13 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
38	33, 37	bitrdi 290	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
39	38	exp32 426	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))))
40	39	com23 87	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))))
41	40	imp 412	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
42	41	pm5.32d 588	. . . . . . . 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 282	. . . . . . 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 3188	. . . . . 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 3198	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
46	44, 45	bitrdi 290	. . . . 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 285	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
48	2, 47	bitrid 286	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
49	48	eqabdv 2899	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦})
50	1, 49	eqtr4id 2820	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744  ∃wrex 3092  Vcvv 3458   ∩ cin 3907  {csn 4594   class class class wbr 5114  ^◡ccnv 5665  ran crn 5667   “ cima 5669   Fn wfn 6538  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552

This theorem is used by:  isoini2  7348  isoselem  7350  infxpenlem  10016