Theorem supisolem 8936

Description: Lemma for supiso 8938. (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 514	. 2 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴))
4		simpll 765	. . . . . . . 8 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5	4	adantr 483	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
6		simplr 767	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝐷 ∈ 𝐴)
7		simplr 767	. . . . . . . 8 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	sselda 3966	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐴)
9		isorel 7078	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐷𝑅𝑦 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
10	5, 6, 8, 9	syl12anc 834	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (𝐷𝑅𝑦 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
11	10	notbid 320	. . . . 5 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (¬ 𝐷𝑅𝑦 ↔ ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
12	11	ralbidva 3196	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
13		isof1o 7075	. . . . . . 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 6615	. . . . . 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 5069	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑦) → ((𝐹‘𝐷)𝑆𝑤 ↔ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
18	17	notbid 320	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (¬ (𝐹‘𝐷)𝑆𝑤 ↔ ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
19	18	ralima 6999	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ↔ ∀𝑦 ∈ 𝐶 ¬ (𝐹‘𝐷)𝑆(𝐹‘𝑦)))
20	16, 7, 19	syl2anc 586	. . . 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 284	. . 3 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ↔ ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤))
22	4	adantr 483	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
23		simpr 487	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
24		simplr 767	. . . . . . 7 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐷 ∈ 𝐴)
25		isorel 7078	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑦𝑅𝐷 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝐷)))
26	22, 23, 24, 25	syl12anc 834	. . . . . 6 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝐷 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝐷)))
27	22	adantr 483	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
28		simplr 767	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐴)
29	7	adantr 483	. . . . . . . . . 10 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
30	29	sselda 3966	. . . . . . . . 9 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
31		isorel 7078	. . . . . . . . 9 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
32	27, 28, 30, 31	syl12anc 834	. . . . . . . 8 ⊢ (((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → (𝑦𝑅𝑧 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
33	32	rexbidva 3296	. . . . . . 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 483	. . . . . . . 8 ⊢ ((((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐷 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
35		breq2 5069	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑧) → ((𝐹‘𝑦)𝑆𝑣 ↔ (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
36	35	rexima 6998	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑧 ∈ 𝐶 (𝐹‘𝑦)𝑆(𝐹‘𝑧)))
37	34, 29, 36	syl2anc 586	. . . . . . 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 284	. . . . . 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 347	. . . . 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 3196	. . . 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 6621	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
42		breq1 5068	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦)𝑆(𝐹‘𝐷) ↔ 𝑤𝑆(𝐹‘𝐷)))
43		breq1 5068	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦)𝑆𝑣 ↔ 𝑤𝑆𝑣))
44	43	rexbidv 3297	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑤 → (∃𝑣 ∈ (𝐹 “ 𝐶)(𝐹‘𝑦)𝑆𝑣 ↔ ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
45	42, 44	imbi12d 347	. . . . . 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 7044	. . . . 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 281	. . 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 632	. 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 582	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 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935   class class class wbr 5065   “ cima 5557   Fn wfn 6349  –onto→wfo 6352  –1-1-onto→wf1o 6353  ‘cfv 6354   Isom wiso 6355

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 5202  ax-nul 5209  ax-pr 5329

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 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363

This theorem is referenced by:  supisoex  8937  supiso  8938