Theorem isoini 6024

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 5128	. 2 ⊢ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦}
2		elin 3412	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
3		isof1o 6013	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1ofo 5646	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
5		forn 5618	. . . . . . . . . 10 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
6	5	eleq2d 2308	. . . . . . . . 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 5640	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
9		fvelrnb 5750	. . . . . . . . 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 190	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦))
12	11	adantr 276	. . . . . 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 340	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 Fn 𝐴 ∧ 𝐷 ∈ 𝐴))
15		funfvex 5712	. . . . . . . 8 ⊢ ((Fun 𝐻 ∧ 𝐷 ∈ dom 𝐻) → (𝐻‘𝐷) ∈ V)
16	15	funfni 5483	. . . . . . 7 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐻‘𝐷) ∈ V)
17		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
18	17	eliniseg 5157	. . . . . . 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 477	. . . . 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 3412	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
22		vex 2824	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
23	22	eliniseg 5157	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
24	23	anbi2d 468	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
25	21, 24	bitrid 192	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷)))
26	25	anbi1d 469	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
27		anass 405	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦)))
28	26, 27	bitrdi 196	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
29	28	adantl 277	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦))))
30		isorel 6014	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
31		fnbrfvb 5741	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
32	31	bicomd 141	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
33	13, 32	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
34	33	adantrr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻‘𝑥) = 𝑦))
35	30, 34	anbi12d 477	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦)))
36		ancom 266	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
37		breq1 4133	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑥) = 𝑦 → ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ↔ 𝑦𝑆(𝐻‘𝐷)))
38	37	pm5.32i 458	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥) = 𝑦 ∧ (𝐻‘𝑥)𝑆(𝐻‘𝐷)) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
39	36, 38	bitri 184	. . . . . . . . . . . . 13 ⊢ (((𝐻‘𝑥)𝑆(𝐻‘𝐷) ∧ (𝐻‘𝑥) = 𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
40	35, 39	bitrdi 196	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
41	40	exp32 365	. . . . . . . . . . 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 124	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ((𝑥𝑅𝐷 ∧ 𝑥𝐻𝑦) ↔ ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))))
44	43	pm5.32d 454	. . . . . . . 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 188	. . . . . . 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 2553	. . . . . 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 2707	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐴 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
48	46, 47	bitrdi 196	. . . . 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 191	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
50	2, 49	bitrid 192	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑦 ∈ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦))
51	50	abbi2dv 2359	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (◡𝑅 “ {𝐷}))𝑥𝐻𝑦})
52	1, 51	eqtr4id 2290	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   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821   ∩ cin 3219  {csn 3709   class class class wbr 4130  ^◡ccnv 4773  ran crn 4775   “ cima 4777   Fn wfn 5372  –onto→wfo 5375  –1-1-onto→wf1o 5376  ‘cfv 5377   Isom wiso 5378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386

This theorem is used by:  isoini2  6025  isoselem  6026