Theorem isomin 7084

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 4308	. . . 4 ⊢ (¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
2		ssel 3960	. . . . . . . . . . . . . 14 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
3		isof1o 7070	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1ofn 6610	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
5		fnbrfvb 6712	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
6	5	ex 415	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn 𝐴 → (𝑥 ∈ 𝐴 → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦)))
7	3, 4, 6	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦)))
8	2, 7	syl9r 78	. . . . . . . . . . . . 13 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))))
9	8	imp31 420	. . . . . . . . . . . 12 ⊢ (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → ((𝐻‘𝑥) = 𝑦 ↔ 𝑥𝐻𝑦))
10	9	rexbidva 3296	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐻𝑦))
11		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
12	11	elima 5928	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐻 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐻𝑦)
13	10, 12	syl6rbbr 292	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐻 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦))
14		fvex 6677	. . . . . . . . . . 11 ⊢ (𝐻‘𝐷) ∈ V
15	11	eliniseg 5952	. . . . . . . . . . 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 632	. . . . . . . . 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 4168	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ (𝑦 ∈ (𝐻 “ 𝐶) ∧ 𝑦 ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
19		r19.41v 3347	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐶 ((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) ↔ (∃𝑥 ∈ 𝐶 (𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)))
20	17, 18, 19	3bitr4g 316	. . . . . . . 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 5061	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑥) = 𝑦 → ((𝐻‘𝑥)𝑆(𝐻‘𝐷) ↔ 𝑦𝑆(𝐻‘𝐷)))
23	22	biimpar 480	. . . . . . . . . . . . 13 ⊢ (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → (𝐻‘𝑥)𝑆(𝐻‘𝐷))
24		vex 3497	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
25	24	eliniseg 5952	. . . . . . . . . . . . . . 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 7073	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
28	26, 27	bitrd 281	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
29	23, 28	syl5ibr 248	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷})))
30	29	exp32 423	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → (((𝐻‘𝑥) = 𝑦 ∧ 𝑦𝑆(𝐻‘𝐷)) → 𝑥 ∈ (◡𝑅 “ {𝐷})))))
31	2, 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 421	. . . . . . . 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 3272	. . . . . . 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 242	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷})))
36		elin 4168	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
37	36	exbii 1844	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
38		neq0 4308	. . . . . . 7 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})))
39		df-rex 3144	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
40	37, 38, 39	3bitr4i 305	. . . . . 6 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 ∈ 𝐶 𝑥 ∈ (◡𝑅 “ {𝐷}))
41	35, 40	syl6ibr 254	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦 ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))
42	41	exlimdv 1930	. . . 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 244	. . 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	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
46		fnfvima 6989	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶))
47	46	3expia 1117	. . . . . . . . . 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 582	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
50	49	adantrd 494	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
51	27	biimpd 231	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
52		fvex 6677	. . . . . . . . . . . . . . . 16 ⊢ (𝐻‘𝑥) ∈ V
53	52	eliniseg 5952	. . . . . . . . . . . . . . 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 254	. . . . . . . . . . . . 13 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
56	26, 55	sylbid 242	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
57	56	exp32 423	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))))
58	2, 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 421	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))
61	60	impd 413	. . . . . . 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 515	. . . . . 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 4168	. . . . . 6 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
64	62, 36, 63	3imtr4g 298	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)}))))
65		n0i 4298	. . . . 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 1930	. . 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 244	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
69	44, 68	impcon4bid 229	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 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  {csn 4560   class class class wbr 5058  ^◡ccnv 5548   “ cima 5552   Fn wfn 6344  –1-1-onto→wf1o 6348  ‘cfv 6349   Isom wiso 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-f1o 6356  df-fv 6357  df-isom 6358

This theorem is referenced by:  isofrlem  7087