Theorem ordthmeolem 21604

Description: Lemma for ordthmeo 21605. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypotheses

Ref	Expression
ordthmeo.1	⊢ 𝑋 = dom 𝑅
ordthmeo.2	⊢ 𝑌 = dom 𝑆

Assertion

Ref	Expression
ordthmeolem	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)))

Proof of Theorem ordthmeolem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isof1o 6573	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) → 𝐹:𝑋–1-1-onto→𝑌)
2	1	3ad2ant3 1084	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌)
3		f1of 6137	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
4	2, 3	syl 17	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋⟶𝑌)
5		fimacnv 6347	. . . . . . 7 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (◡𝐹 “ 𝑌) = 𝑋)
7		ordthmeo.1	. . . . . . . . 9 ⊢ 𝑋 = dom 𝑅
8	7	ordttopon 20997	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
9	8	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
10		toponmax 20730	. . . . . . 7 ⊢ ((ordTop‘𝑅) ∈ (TopOn‘𝑋) → 𝑋 ∈ (ordTop‘𝑅))
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑋 ∈ (ordTop‘𝑅))
12	6, 11	eqeltrd 2701	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (◡𝐹 “ 𝑌) ∈ (ordTop‘𝑅))
13		elsni 4194	. . . . . . 7 ⊢ (𝑧 ∈ {𝑌} → 𝑧 = 𝑌)
14	13	imaeq2d 5466	. . . . . 6 ⊢ (𝑧 ∈ {𝑌} → (◡𝐹 “ 𝑧) = (◡𝐹 “ 𝑌))
15	14	eleq1d 2686	. . . . 5 ⊢ (𝑧 ∈ {𝑌} → ((◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ (◡𝐹 “ 𝑌) ∈ (ordTop‘𝑅)))
16	12, 15	syl5ibrcom 237	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝑧 ∈ {𝑌} → (◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅)))
17	16	ralrimiv 2965	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ {𝑌} (◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅))
18		cnvimass 5485	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ dom 𝐹
19		f1odm 6141	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → dom 𝐹 = 𝑋)
20	2, 19	syl 17	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → dom 𝐹 = 𝑋)
21	20	adantr 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → dom 𝐹 = 𝑋)
22	18, 21	syl5sseq 3653	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋)
23		sseqin2 3817	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋 ↔ (𝑋 ∩ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}))
24	22, 23	sylib 208	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 ∩ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}))
25	2	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → 𝐹:𝑋–1-1-onto→𝑌)
26		f1ofn 6138	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋)
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → 𝐹 Fn 𝑋)
28		elpreima 6337	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})))
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})))
30		simpr 477	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
31	30	biantrurd 529	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})))
32	4	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋⟶𝑌)
33	32	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ 𝑌)
34		breq1 4656	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑧) → (𝑦𝑆𝑥 ↔ (𝐹‘𝑧)𝑆𝑥))
35	34	notbid 308	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑧) → (¬ 𝑦𝑆𝑥 ↔ ¬ (𝐹‘𝑧)𝑆𝑥))
36	35	elrab3 3364	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ 𝑌 → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ (𝐹‘𝑧)𝑆𝑥))
37	33, 36	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ (𝐹‘𝑧)𝑆𝑥))
38		simpll3 1102	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌))
39		f1ocnv 6149	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
40		f1of 6137	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
41	2, 39, 40	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ◡𝐹:𝑌⟶𝑋)
42	41	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
43	42	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (◡𝐹‘𝑥) ∈ 𝑋)
44		isorel 6576	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ (◡𝐹‘𝑥) ∈ 𝑋)) → (𝑧𝑅(◡𝐹‘𝑥) ↔ (𝐹‘𝑧)𝑆(𝐹‘(◡𝐹‘𝑥))))
45	38, 30, 43, 44	syl12anc 1324	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑧𝑅(◡𝐹‘𝑥) ↔ (𝐹‘𝑧)𝑆(𝐹‘(◡𝐹‘𝑥))))
46		f1ocnvfv2 6533	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
47	2, 46	sylan 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
48	47	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
49	48	breq2d 4665	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧)𝑆(𝐹‘(◡𝐹‘𝑥)) ↔ (𝐹‘𝑧)𝑆𝑥))
50	45, 49	bitrd 268	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑧𝑅(◡𝐹‘𝑥) ↔ (𝐹‘𝑧)𝑆𝑥))
51	50	notbid 308	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑧𝑅(◡𝐹‘𝑥) ↔ ¬ (𝐹‘𝑧)𝑆𝑥))
52	37, 51	bitr4d 271	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ 𝑧𝑅(◡𝐹‘𝑥)))
53	29, 31, 52	3bitr2d 296	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ ¬ 𝑧𝑅(◡𝐹‘𝑥)))
54	53	rabbi2dva 3821	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 ∩ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})) = {𝑧 ∈ 𝑋 ∣ ¬ 𝑧𝑅(◡𝐹‘𝑥)})
55	24, 54	eqtr3d 2658	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) = {𝑧 ∈ 𝑋 ∣ ¬ 𝑧𝑅(◡𝐹‘𝑥)})
56		simpl1 1064	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ 𝑉)
57	7	ordtopn1 20998	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (◡𝐹‘𝑥) ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ¬ 𝑧𝑅(◡𝐹‘𝑥)} ∈ (ordTop‘𝑅))
58	56, 42, 57	syl2anc 693	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → {𝑧 ∈ 𝑋 ∣ ¬ 𝑧𝑅(◡𝐹‘𝑥)} ∈ (ordTop‘𝑅))
59	55, 58	eqeltrd 2701	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
60	59	ralrimiva 2966	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥 ∈ 𝑌 (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
61		ordthmeo.2	. . . . . . . . . 10 ⊢ 𝑌 = dom 𝑆
62		dmexg 7097	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑊 → dom 𝑆 ∈ V)
63	61, 62	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑊 → 𝑌 ∈ V)
64	63	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑌 ∈ V)
65		rabexg 4812	. . . . . . . 8 ⊢ (𝑌 ∈ V → {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
66	64, 65	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
67	66	ralrimivw 2967	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥 ∈ 𝑌 {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
68		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) = (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})
69		imaeq2 5462	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} → (◡𝐹 “ 𝑧) = (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}))
70	69	eleq1d 2686	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} → ((◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
71	68, 70	ralrnmpt 6368	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑌 {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V → (∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥 ∈ 𝑌 (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
72	67, 71	syl 17	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥 ∈ 𝑌 (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
73	60, 72	mpbird 247	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅))
74		cnvimass 5485	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ dom 𝐹
75	74, 21	syl5sseq 3653	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋)
76		sseqin2 3817	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋 ↔ (𝑋 ∩ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))
77	75, 76	sylib 208	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 ∩ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))
78		elpreima 6337	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})))
79	27, 78	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})))
80	30	biantrurd 529	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})))
81		breq2 4657	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑧) → (𝑥𝑆𝑦 ↔ 𝑥𝑆(𝐹‘𝑧)))
82	81	notbid 308	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑧) → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑥𝑆(𝐹‘𝑧)))
83	82	elrab3 3364	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ 𝑌 → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ 𝑥𝑆(𝐹‘𝑧)))
84	33, 83	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ 𝑥𝑆(𝐹‘𝑧)))
85		isorel 6576	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ ((◡𝐹‘𝑥) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((◡𝐹‘𝑥)𝑅𝑧 ↔ (𝐹‘(◡𝐹‘𝑥))𝑆(𝐹‘𝑧)))
86	38, 43, 30, 85	syl12anc 1324	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((◡𝐹‘𝑥)𝑅𝑧 ↔ (𝐹‘(◡𝐹‘𝑥))𝑆(𝐹‘𝑧)))
87	48	breq1d 4663	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘(◡𝐹‘𝑥))𝑆(𝐹‘𝑧) ↔ 𝑥𝑆(𝐹‘𝑧)))
88	86, 87	bitrd 268	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((◡𝐹‘𝑥)𝑅𝑧 ↔ 𝑥𝑆(𝐹‘𝑧)))
89	88	notbid 308	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (¬ (◡𝐹‘𝑥)𝑅𝑧 ↔ ¬ 𝑥𝑆(𝐹‘𝑧)))
90	84, 89	bitr4d 271	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ (◡𝐹‘𝑥)𝑅𝑧))
91	79, 80, 90	3bitr2d 296	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ ¬ (◡𝐹‘𝑥)𝑅𝑧))
92	91	rabbi2dva 3821	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 ∩ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})) = {𝑧 ∈ 𝑋 ∣ ¬ (◡𝐹‘𝑥)𝑅𝑧})
93	77, 92	eqtr3d 2658	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) = {𝑧 ∈ 𝑋 ∣ ¬ (◡𝐹‘𝑥)𝑅𝑧})
94	7	ordtopn2 20999	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (◡𝐹‘𝑥) ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ¬ (◡𝐹‘𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
95	56, 42, 94	syl2anc 693	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → {𝑧 ∈ 𝑋 ∣ ¬ (◡𝐹‘𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
96	93, 95	eqeltrd 2701	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
97	96	ralrimiva 2966	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥 ∈ 𝑌 (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
98		rabexg 4812	. . . . . . . 8 ⊢ (𝑌 ∈ V → {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
99	64, 98	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
100	99	ralrimivw 2967	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥 ∈ 𝑌 {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
101		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) = (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})
102		imaeq2 5462	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} → (◡𝐹 “ 𝑧) = (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))
103	102	eleq1d 2686	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} → ((◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
104	101, 103	ralrnmpt 6368	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑌 {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V → (∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥 ∈ 𝑌 (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
105	100, 104	syl 17	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥 ∈ 𝑌 (◡𝐹 “ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
106	97, 105	mpbird 247	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅))
107		ralunb 3794	. . . 4 ⊢ (∀𝑧 ∈ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ (∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅)))
108	73, 106, 107	sylanbrc 698	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅))
109		ralunb 3794	. . 3 ⊢ (∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})))(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ↔ (∀𝑧 ∈ {𝑌} (◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅) ∧ ∀𝑧 ∈ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅)))
110	17, 108, 109	sylanbrc 698	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})))(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅))
111		eqid 2622	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) = ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥})
112		eqid 2622	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}) = ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})
113	61, 111, 112	ordtuni 20994	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → 𝑌 = ∪ ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))))
114	113, 63	eqeltrrd 2702	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → ∪ ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
115		uniexb 6973	. . . . 5 ⊢ (({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V ↔ ∪ ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
116	114, 115	sylibr 224	. . . 4 ⊢ (𝑆 ∈ 𝑊 → ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
117	116	3ad2ant2 1083	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
118	61, 111, 112	ordtval 20993	. . . 4 ⊢ (𝑆 ∈ 𝑊 → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
119	118	3ad2ant2 1083	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
120	61	ordttopon 20997	. . . 4 ⊢ (𝑆 ∈ 𝑊 → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
121	120	3ad2ant2 1083	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
122	9, 117, 119, 121	subbascn 21058	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥 ∈ 𝑌 ↦ {𝑦 ∈ 𝑌 ∣ ¬ 𝑥𝑆𝑦})))(◡𝐹 “ 𝑧) ∈ (ordTop‘𝑅))))
123	4, 110, 122	mpbir2and 957	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650  ficfi 8316  topGenctg 16098  ordTopcordt 16159  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-topgen 16104  df-ordt 16161  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031

This theorem is referenced by:  ordthmeo  21605  xrmulc1cn  29976