Theorem isomin 7208

Description: Isomorphisms preserve minimal elements. Note that (^◡𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥 ∣ 𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)

Assertion

Ref	Expression
isomin	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))

Proof of Theorem isomin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4279	. . . 4 ⊢ (¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
2		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
3	2	elima 5974	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐻 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐻𝑦)
4		ssel 3914	. . . . . . . . . . . . . 14 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
5		isof1o 7194	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
6		f1ofn 6717	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
7		fnbrfvb 6822	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
8	7	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn 𝐴 → (𝑥 ∈ 𝐴 → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦)))
9	5, 6, 8	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦)))
10	4, 9	syl9r 78	. . . . . . . . . . . . 13 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))))
11	10	imp31 418	. . . . . . . . . . . 12 ⊢ (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
12	11	rexbidva 3225	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐻𝑦))
13	3, 12	bitr4id 290	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐻 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦))
14		fvex 6787	. . . . . . . . . . 11 ⊢ (𝐻‘𝐷) ∈ V
15	2	eliniseg 6002	. . . . . . . . . . 11 ⊢ ((𝐻‘𝐷) ∈ V → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
16	14, 15	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ 𝑦𝑆(𝐻‘𝐷)))
17	13, 16	anbi12d 631	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → ((𝑦 ∈ (𝐻 “ 𝐶) ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
18		elin 3903	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ (𝐻 “ 𝐶) ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
19		r19.41v 3276	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐶 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
20	17, 18, 19	3bitr4g 314	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ 𝐶 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
21	20	adantrr 714	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ∃𝑥 ∈ 𝐶 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷))))
22		breq1 5077	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑥) = 𝑦 → ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ↔ 𝑦𝑆(𝐻‘𝐷)))
23	22	biimpar 478	. . . . . . . . . . . . 13 ⊢ (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → (𝐻‘𝑥)𝑆(𝐻‘𝐷))
24		vex 3436	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
25	24	eliniseg 6002	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
26	25	ad2antll 726	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27		isorel 7197	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
28	26, 27	bitrd 278	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
29	23, 28	syl5ibr 245	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷})))
30	29	exp32 421	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷})))))
31	4, 30	syl9r 78	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → (𝐷 ∈ 𝐴 → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷}))))))
32	31	com34 91	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐶 → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷}))))))
33	32	imp32 419	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷}))))
34	33	reximdvai 3200	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∃𝑥 ∈ 𝐶 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → ∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷})))
35	21, 34	sylbid 239	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷})))
36		elin 3903	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
37	36	exbii 1850	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
38		neq0 4279	. . . . . . 7 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})))
39		df-rex 3070	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
40	37, 38, 39	3bitr4i 303	. . . . . 6 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷}))
41	35, 40	syl6ibr 251	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))
42	41	exlimdv 1936	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∃𝑦 𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))
43	1, 42	syl5bi 241	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ → ¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))
44	43	con4d 115	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ → ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
45	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
46		fnfvima 7109	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶))
47	46	3expia 1120	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
48	47	adantrr 714	. . . . . . . . 9 ⊢ ((𝐻 Fn 𝐴 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
49	45, 48	sylan 580	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
50	49	adantrd 492	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
51	27	biimpd 228	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
52		fvex 6787	. . . . . . . . . . . . . . . 16 ⊢ (𝐻‘𝑥) ∈ V
53	52	eliniseg 6002	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝐷) ∈ V → ((𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
54	14, 53	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷))
55	51, 54	syl6ibr 251	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
56	26, 55	sylbid 239	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
57	56	exp32 421	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))))
58	4, 57	syl9r 78	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))))
59	58	com34 91	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))))
60	59	imp32 419	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))
61	60	impd 411	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
62	50, 61	jcad 513	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))
63		elin 3903	. . . . . 6 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
64	62, 36, 63	3imtr4g 296	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)}))))
65		n0i 4267	. . . . 5 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅)
66	64, 65	syl6 35	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
67	66	exlimdv 1936	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
68	38, 67	syl5bi 241	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
69	44, 68	impcon4bid 226	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561   class class class wbr 5074  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  –1-1-onto→wf1o 6432  ‘cfv 6433   Isom wiso 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-f1o 6440  df-fv 6441  df-isom 6442

This theorem is referenced by:  isofrlem  7211