Theorem supisolem 6997

Description: Lemma for supisoti 6999. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supiso.1	⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
supiso.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
supisolem	⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))

Distinct variable groups:   𝑤,𝑣,𝑦,𝑧,𝐴   𝑣,𝐶,𝑤,𝑦,𝑧   𝑤,𝐷,𝑦,𝑧   𝜑,𝑤   𝑣,𝐹,𝑤,𝑦,𝑧   𝑤,𝑅,𝑦,𝑧   𝑣,𝑆,𝑤,𝑦,𝑧   𝑣,𝐵,𝑤,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣)   𝐷(𝑣)   𝑅(𝑣)

Proof of Theorem supisolem

Step	Hyp	Ref	Expression
1		supiso.1	. . 3 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		supiso.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3	1, 2	jca 306	. 2 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴))
4		simpll 527	. . . . . . . 8 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5	4	adantr 276	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
6		simplr 528	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝐷 ∈ 𝐴)
7		simplr 528	. . . . . . . 8 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	sselda 3153	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐴)
9		isorel 5799	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐷𝑅𝑦 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
10	5, 6, 8, 9	syl12anc 1236	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (𝐷𝑅𝑦 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
11	10	notbid 667	. . . . 5 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (¬ 𝐷𝑅𝑦 ↔ ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
12	11	ralbidva 2471	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
13		isof1o 5798	. . . . . . 7 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
14	4, 13	syl 14	. . . . . 6 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹:𝐴–1-1-onto→𝐵)
15		f1ofn 5454	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
16	14, 15	syl 14	. . . . 5 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹 Fn 𝐴)
17		breq2 4002	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑦) → ((𝐹‘𝐷)𝑆𝑤 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
18	17	notbid 667	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (¬ (𝐹‘𝐷)𝑆𝑤 ↔ ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
19	18	ralima 5747	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
20	16, 7, 19	syl2anc 411	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
21	12, 20	bitr4d 191	. . 3 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ↔ ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤))
22	4	adantr 276	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
23		simpr 110	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
24		simplr 528	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐷 ∈ 𝐴)
25		isorel 5799	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦𝑅𝐷 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝐷)))
26	22, 23, 24, 25	syl12anc 1236	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝐷 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝐷)))
27	22	adantr 276	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
28		simplr 528	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐴)
29	7	adantr 276	. . . . . . . . . 10 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
30	29	sselda 3153	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
31		isorel 5799	. . . . . . . . 9 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
32	27, 28, 30, 31	syl12anc 1236	. . . . . . . 8 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → (𝑦𝑅𝑧 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
33	32	rexbidva 2472	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝐶 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐶 (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
34	16	adantr 276	. . . . . . . 8 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
35		breq2 4002	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑧) → ((𝐹‘𝑦)𝑆𝑣 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
36	35	rexima 5746	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑧 ∈ 𝐶 (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
37	34, 29, 36	syl2anc 411	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑧 ∈ 𝐶 (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
38	33, 37	bitr4d 191	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝐶 𝑦𝑅𝑧 ↔ ∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣))
39	26, 38	imbi12d 234	. . . . 5 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧) ↔ ((𝐹‘𝑦)𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣)))
40	39	ralbidva 2471	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦)𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣)))
41		f1ofo 5460	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
42		breq1 4001	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦)𝑆(𝐹‘𝐷) ↔ 𝑤𝑆(𝐹‘𝐷)))
43		breq1 4001	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦)𝑆𝑣 ↔ 𝑤𝑆𝑣))
44	43	rexbidv 2476	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑤 → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
45	42, 44	imbi12d 234	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑤 → (((𝐹‘𝑦)𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣) ↔ (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
46	45	cbvfo 5776	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦)𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣) ↔ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
47	14, 41, 46	3syl 17	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦)𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣) ↔ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
48	40, 47	bitrd 188	. . 3 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧) ↔ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
49	21, 48	anbi12d 473	. 2 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
50	3, 49	sylan 283	1 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454   ⊆ wss 3127   class class class wbr 3998   “ cima 4623   Fn wfn 5203  –onto→wfo 5206  –1-1-onto→wf1o 5207  ‘cfv 5208   Isom wiso 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217

This theorem is referenced by:  supisoex  6998  supisoti  6999