Theorem isomin 7330

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 4344	. . . 4 ⊢ (¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
2		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
3	2	elima 6062	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐻 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐻𝑦)
4		ssel 3974	. . . . . . . . . . . . . 14 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
5		isof1o 7316	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
6		f1ofn 6831	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
7		fnbrfvb 6941	. . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐻𝑦))
13	3, 12	bitr4id 289	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐻 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦))
14		fvex 6901	. . . . . . . . . . 11 ⊢ (𝐻‘𝐷) ∈ V
15	2	eliniseg 6090	. . . . . . . . . . 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 3963	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ (𝐻 “ 𝐶) ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
19		r19.41v 3188	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐶 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
20	17, 18, 19	3bitr4g 313	. . . . . . . 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 715	. . . . . . 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 5150	. . . . . . . . . . . . . 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 3478	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
25	24	eliniseg 6090	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
26	25	ad2antll 727	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27		isorel 7319	. . . . . . . . . . . . . 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	imbitrrid 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 3165	. . . . . . 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 3963	. . . . . . . 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 4344	. . . . . . 7 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})))
39		df-rex 3071	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
40	37, 38, 39	3bitr4i 302	. . . . . 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	biimtrid 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 7231	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶))
47	46	3expia 1121	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
48	47	adantrr 715	. . . . . . . . 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 6901	. . . . . . . . . . . . . . . 16 ⊢ (𝐻‘𝑥) ∈ V
53	52	eliniseg 6090	. . . . . . . . . . . . . . 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 3963	. . . . . 6 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
64	62, 36, 63	3imtr4g 295	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)}))))
65		n0i 4332	. . . . 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	biimtrid 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 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627   class class class wbr 5147  ^◡ccnv 5674   “ cima 5678   Fn wfn 6535  –1-1-onto→wf1o 6539  ‘cfv 6540   Isom wiso 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-f1o 6547  df-fv 6548  df-isom 6549

This theorem is referenced by:  isofrlem  7333