Theorem supisolem 8648

Description: Lemma for supiso 8650. (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 507	. 2 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴))
4		simpll 783	. . . . . . . 8 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5	4	adantr 474	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
6		simplr 785	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝐷 ∈ 𝐴)
7		simplr 785	. . . . . . . 8 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	sselda 3827	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐴)
9		isorel 6831	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐷𝑅𝑦 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
10	5, 6, 8, 9	syl12anc 870	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (𝐷𝑅𝑦 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
11	10	notbid 310	. . . . 5 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (¬ 𝐷𝑅𝑦 ↔ ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
12	11	ralbidva 3194	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
13		isof1o 6828	. . . . . . 7 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
14	4, 13	syl 17	. . . . . 6 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹:𝐴–1-1-onto→𝐵)
15		f1ofn 6379	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
16	14, 15	syl 17	. . . . 5 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹 Fn 𝐴)
17		breq2 4877	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑦) → ((𝐹‘𝐷)𝑆𝑤 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
18	17	notbid 310	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (¬ (𝐹‘𝐷)𝑆𝑤 ↔ ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
19	18	ralima 6754	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
20	16, 7, 19	syl2anc 579	. . . 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 274	. . 3 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ↔ ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤))
22	4	adantr 474	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
23		simpr 479	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
24		simplr 785	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐷 ∈ 𝐴)
25		isorel 6831	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦𝑅𝐷 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝐷)))
26	22, 23, 24, 25	syl12anc 870	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝐷 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝐷)))
27	22	adantr 474	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
28		simplr 785	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐴)
29	7	adantr 474	. . . . . . . . . 10 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
30	29	sselda 3827	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
31		isorel 6831	. . . . . . . . 9 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
32	27, 28, 30, 31	syl12anc 870	. . . . . . . 8 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → (𝑦𝑅𝑧 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
33	32	rexbidva 3259	. . . . . . 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 474	. . . . . . . 8 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
35		breq2 4877	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑧) → ((𝐹‘𝑦)𝑆𝑣 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
36	35	rexima 6753	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑧 ∈ 𝐶 (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
37	34, 29, 36	syl2anc 579	. . . . . . 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 274	. . . . . 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 336	. . . . 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 3194	. . . 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 6385	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
42		breq1 4876	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦)𝑆(𝐹‘𝐷) ↔ 𝑤𝑆(𝐹‘𝐷)))
43		breq1 4876	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦)𝑆𝑣 ↔ 𝑤𝑆𝑣))
44	43	rexbidv 3262	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑤 → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
45	42, 44	imbi12d 336	. . . . . 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 6799	. . . . 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 18	. . . 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 271	. . 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 624	. 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 575	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   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118   ⊆ wss 3798   class class class wbr 4873   “ cima 5345   Fn wfn 6118  –onto→wfo 6121  –1-1-onto→wf1o 6122  ‘cfv 6123   Isom wiso 6124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132

This theorem is referenced by:  supisoex  8649  supiso  8650