Theorem soisores 7336

Description: Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)

Assertion

Ref	Expression
soisores	⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦)

Proof of Theorem soisores

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isorel 7335	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦)))
2		fvres 6907	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
3		fvres 6907	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
4	2, 3	breqan12d 5130	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
5	4	adantl 487	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
6	1, 5	bitrd 282	. . . 4 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
7	6	biimpd 232	. . 3 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
8	7	ralrimivva 3211	. 2 ⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
9		ffn 6712	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵)
10	9	ad2antrl 741	. . . . . . 7 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → 𝐹 Fn 𝐵)
11		simprr 785	. . . . . . 7 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ⊆ 𝐵)
12		fnssres 6665	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	syl2anc 596	. . . . . 6 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → (𝐹 ↾ 𝐴) Fn 𝐴)
14	13	3adant3 1150	. . . . 5 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝐹 ↾ 𝐴) Fn 𝐴)
15		df-ima 5679	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
16	15	eqcomi 2775	. . . . . 6 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
17	16	a1i 11	. . . . 5 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴))
18		fvres 6907	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑧) = (𝐹‘𝑧))
19		fvres 6907	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑤) = (𝐹‘𝑤))
20	18, 19	eqeqan12d 2780	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
21	20	adantl 487	. . . . . . 7 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
22		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
23		simprr 785	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
24		simpl3 1212	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
25		breq1 5117	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
26		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
27	26	breq1d 5124	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
28	25, 27	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑧𝑅𝑦 → (𝐹‘𝑧)𝑆(𝐹‘𝑦))))
29		breq2 5118	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
30		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
31	30	breq2d 5126	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
32	29, 31	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑧𝑅𝑦 → (𝐹‘𝑧)𝑆(𝐹‘𝑦)) ↔ (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤))))
33	28, 32	rspc2va 3596	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
34	22, 23, 24, 33	syl21anc 851	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
35		breq1 5117	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑤𝑅𝑦))
36		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
37	36	breq1d 5124	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑦)))
38	35, 37	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑤𝑅𝑦 → (𝐹‘𝑤)𝑆(𝐹‘𝑦))))
39		breq2 5118	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤𝑅𝑦 ↔ 𝑤𝑅𝑧))
40		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
41	40	breq2d 5126	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
42	39, 41	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑤𝑅𝑦 → (𝐹‘𝑤)𝑆(𝐹‘𝑦)) ↔ (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
43	38, 42	rspc2va 3596	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
44	23, 22, 24, 43	syl21anc 851	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
45	34, 44	orim12d 979	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
46	45	con3d 153	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧)) → ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
47		simpl1r 1244	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑆 Or 𝐶)
48		simpl2l 1245	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝐹:𝐵⟶𝐶)
49		simpl2r 1246	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
50	49, 22	sseldd 3941	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑧 ∈ 𝐵)
51	48, 50	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑧) ∈ 𝐶)
52	49, 23	sseldd 3941	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐵)
53	48, 52	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐶)
54		sotrieq 5605	. . . . . . . . 9 ⊢ ((𝑆 Or 𝐶 ∧ ((𝐹‘𝑧) ∈ 𝐶 ∧ (𝐹‘𝑤) ∈ 𝐶)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
55	47, 51, 53, 54	syl12anc 850	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
56		simpl1l 1243	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑅 Or 𝐵)
57		sotrieq 5605	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
58	56, 50, 52, 57	syl12anc 850	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
59	46, 55, 58	3imtr4d 297	. . . . . . 7 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
60	21, 59	sylbid 243	. . . . . 6 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) → 𝑧 = 𝑤))
61	60	ralrimivva 3211	. . . . 5 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) → 𝑧 = 𝑤))
62		dff1o6 7284	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴) ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) → 𝑧 = 𝑤)))
63	14, 17, 61, 62	syl3anbrc 1362	. . . 4 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
64		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤)))
66	65, 44	orim12d 979	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑧 = 𝑤 ∨ 𝑤𝑅𝑧) → ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
67	66	con3d 153	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧)) → ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
68		sotric 5604	. . . . . . . . 9 ⊢ ((𝑆 Or 𝐶 ∧ ((𝐹‘𝑧) ∈ 𝐶 ∧ (𝐹‘𝑤) ∈ 𝐶)) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
69	47, 51, 53, 68	syl12anc 850	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
70		sotric 5604	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
71	56, 50, 52, 70	syl12anc 850	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
72	67, 69, 71	3imtr4d 297	. . . . . . 7 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) → 𝑧𝑅𝑤))
73	34, 72	impbid 215	. . . . . 6 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
74	18, 19	breqan12d 5130	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
75	74	adantl 487	. . . . . 6 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
76	73, 75	bitr4d 285	. . . . 5 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 ↔ ((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤)))
77	76	ralrimivva 3211	. . . 4 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ ((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤)))
78		df-isom 6552	. . . 4 ⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ ((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤))))
79	63, 77, 78	sylanbrc 595	. . 3 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)))
80	79	3expia 1139	. 2 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) → (𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴))))
81	8, 80	impbid2 229	1 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ⊆ wss 3908   class class class wbr 5114   Or wor 5573  ran crn 5667   ↾ cres 5668   “ cima 5669   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-po 5574  df-so 5575  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552

This theorem is used by:  isercolllem1  15742  dvgt0lem2  26199  erdszelem4  35707  erdszelem8  35711  erdsze2lem2  35717