Theorem soisores 6477

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 6476	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦)))
2		fvres 6117	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
3		fvres 6117	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
4	2, 3	breqan12d 4599	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
5	4	adantl 481	. . . . 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 267	. . . 4 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
7	6	biimpd 218	. . 3 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
8	7	ralrimivva 2954	. 2 ⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
9		ffn 5958	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵)
10	9	ad2antrl 760	. . . . . . 7 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → 𝐹 Fn 𝐵)
11		simprr 792	. . . . . . 7 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ⊆ 𝐵)
12		fnssres 5918	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	syl2anc 691	. . . . . 6 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → (𝐹 ↾ 𝐴) Fn 𝐴)
14	13	3adant3 1074	. . . . 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 5051	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
16	15	eqcomi 2619	. . . . . 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 6117	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑧) = (𝐹‘𝑧))
19		fvres 6117	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑤) = (𝐹‘𝑤))
20	18, 19	eqeqan12d 2626	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
21	20	adantl 481	. . . . . . 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 790	. . . . . . . . . . 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 792	. . . . . . . . . . 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 1059	. . . . . . . . . . 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 4586	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
26		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
27	26	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
28	25, 27	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑧𝑅𝑦 → (𝐹‘𝑧)𝑆(𝐹‘𝑦))))
29		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
30		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
31	30	breq2d 4595	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
32	29, 31	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑧𝑅𝑦 → (𝐹‘𝑧)𝑆(𝐹‘𝑦)) ↔ (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤))))
33	28, 32	rspc2va 3294	. . . . . . . . . . 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 1317	. . . . . . . . . 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 4586	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑤𝑅𝑦))
36		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
37	36	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑦)))
38	35, 37	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑤𝑅𝑦 → (𝐹‘𝑤)𝑆(𝐹‘𝑦))))
39		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤𝑅𝑦 ↔ 𝑤𝑅𝑧))
40		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
41	40	breq2d 4595	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
42	39, 41	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑤𝑅𝑦 → (𝐹‘𝑤)𝑆(𝐹‘𝑦)) ↔ (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
43	38, 42	rspc2va 3294	. . . . . . . . . . 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 1317	. . . . . . . . . 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 879	. . . . . . . . 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 147	. . . . . . . 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 1106	. . . . . . . . 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 1107	. . . . . . . . . 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 1108	. . . . . . . . . . 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 3569	. . . . . . . . . 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	ffvelrnd 6268	. . . . . . . . 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 3569	. . . . . . . . . 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	ffvelrnd 6268	. . . . . . . . 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 4986	. . . . . . . . 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 1316	. . . . . . . 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 1105	. . . . . . . . 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 4986	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
58	56, 50, 52, 57	syl12anc 1316	. . . . . . . 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 282	. . . . . . 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 229	. . . . . 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 2954	. . . . 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 6431	. . . . 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 1239	. . . 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 6103	. . . . . . . . . . 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 879	. . . . . . . . 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 147	. . . . . . . 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 4985	. . . . . . . . 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 1316	. . . . . . . 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 4985	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
71	56, 50, 52, 70	syl12anc 1316	. . . . . . . 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 282	. . . . . . 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 201	. . . . . 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 4599	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
75	74	adantl 481	. . . . . 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 270	. . . . 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 2954	. . . 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 5813	. . . 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 695	. . 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 1259	. 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 215	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))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   class class class wbr 4583   Or wor 4958  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813

This theorem is referenced by:  isercolllem1  14243  dvgt0lem2  23570  erdszelem4  30430  erdszelem8  30434  erdsze2lem2  30440