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Theorem ordthmeolem 21652
Description: Lemma for ordthmeo 21653. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypotheses
Ref Expression
ordthmeo.1 𝑋 = dom 𝑅
ordthmeo.2 𝑌 = dom 𝑆
Assertion
Ref Expression
ordthmeolem ((𝑅𝑉𝑆𝑊𝐹 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.
StepHypRef Expression
1 isof1o 6613 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) → 𝐹:𝑋1-1-onto𝑌)
213ad2ant3 1104 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3 f1of 6175 . . 3 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
42, 3syl 17 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋𝑌)
5 fimacnv 6387 . . . . . . 7 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
64, 5syl 17 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹𝑌) = 𝑋)
7 ordthmeo.1 . . . . . . . . 9 𝑋 = dom 𝑅
87ordttopon 21045 . . . . . . . 8 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
983ad2ant1 1102 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
10 toponmax 20778 . . . . . . 7 ((ordTop‘𝑅) ∈ (TopOn‘𝑋) → 𝑋 ∈ (ordTop‘𝑅))
119, 10syl 17 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑋 ∈ (ordTop‘𝑅))
126, 11eqeltrd 2730 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹𝑌) ∈ (ordTop‘𝑅))
13 elsni 4227 . . . . . . 7 (𝑧 ∈ {𝑌} → 𝑧 = 𝑌)
1413imaeq2d 5501 . . . . . 6 (𝑧 ∈ {𝑌} → (𝐹𝑧) = (𝐹𝑌))
1514eleq1d 2715 . . . . 5 (𝑧 ∈ {𝑌} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹𝑌) ∈ (ordTop‘𝑅)))
1612, 15syl5ibrcom 237 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝑧 ∈ {𝑌} → (𝐹𝑧) ∈ (ordTop‘𝑅)))
1716ralrimiv 2994 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ {𝑌} (𝐹𝑧) ∈ (ordTop‘𝑅))
18 cnvimass 5520 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ dom 𝐹
19 f1odm 6179 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌 → dom 𝐹 = 𝑋)
202, 19syl 17 . . . . . . . . . . 11 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → dom 𝐹 = 𝑋)
2120adantr 480 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → dom 𝐹 = 𝑋)
2218, 21syl5sseq 3686 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋)
23 sseqin2 3850 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
2422, 23sylib 208 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
252ad2antrr 762 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹:𝑋1-1-onto𝑌)
26 f1ofn 6176 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌𝐹 Fn 𝑋)
2725, 26syl 17 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Fn 𝑋)
28 elpreima 6377 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})))
2927, 28syl 17 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})))
30 simpr 476 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝑧𝑋)
3130biantrurd 528 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})))
324adantr 480 . . . . . . . . . . . . 13 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝐹:𝑋𝑌)
3332ffvelrnda 6399 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ 𝑌)
34 breq1 4688 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑦𝑆𝑥 ↔ (𝐹𝑧)𝑆𝑥))
3534notbid 307 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑦𝑆𝑥 ↔ ¬ (𝐹𝑧)𝑆𝑥))
3635elrab3 3397 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ 𝑌 → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ (𝐹𝑧)𝑆𝑥))
3733, 36syl 17 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ (𝐹𝑧)𝑆𝑥))
38 simpll3 1122 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌))
39 f1ocnv 6187 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
40 f1of 6175 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
412, 39, 403syl 18 . . . . . . . . . . . . . . . 16 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑌𝑋)
4241ffvelrnda 6399 . . . . . . . . . . . . . . 15 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
4342adantr 480 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑥) ∈ 𝑋)
44 isorel 6616 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ (𝑧𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
4538, 30, 43, 44syl12anc 1364 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
46 f1ocnvfv2 6573 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
472, 46sylan 487 . . . . . . . . . . . . . . 15 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
4847adantr 480 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹‘(𝐹𝑥)) = 𝑥)
4948breq2d 4697 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧)𝑆(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑧)𝑆𝑥))
5045, 49bitrd 268 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆𝑥))
5150notbid 307 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ 𝑧𝑅(𝐹𝑥) ↔ ¬ (𝐹𝑧)𝑆𝑥))
5237, 51bitr4d 271 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5329, 31, 523bitr2d 296 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5453rabbi2dva 3854 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
5524, 54eqtr3d 2687 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
56 simpl1 1084 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝑅𝑉)
577ordtopn1 21046 . . . . . . . 8 ((𝑅𝑉 ∧ (𝐹𝑥) ∈ 𝑋) → {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)} ∈ (ordTop‘𝑅))
5856, 42, 57syl2anc 694 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)} ∈ (ordTop‘𝑅))
5955, 58eqeltrd 2730 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
6059ralrimiva 2995 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
61 ordthmeo.2 . . . . . . . . . 10 𝑌 = dom 𝑆
62 dmexg 7139 . . . . . . . . . 10 (𝑆𝑊 → dom 𝑆 ∈ V)
6361, 62syl5eqel 2734 . . . . . . . . 9 (𝑆𝑊𝑌 ∈ V)
64633ad2ant2 1103 . . . . . . . 8 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑌 ∈ V)
65 rabexg 4844 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6664, 65syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6766ralrimivw 2996 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
68 eqid 2651 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
69 imaeq2 5497 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
7069eleq1d 2715 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
7168, 70ralrnmpt 6408 . . . . . 6 (∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V → (∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
7267, 71syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
7360, 72mpbird 247 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅))
74 cnvimass 5520 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ dom 𝐹
7574, 21syl5sseq 3686 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋)
76 sseqin2 3850 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
7775, 76sylib 208 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
78 elpreima 6377 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))
7927, 78syl 17 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))
8030biantrurd 528 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))
81 breq2 4689 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑥𝑆𝑦𝑥𝑆(𝐹𝑧)))
8281notbid 307 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
8382elrab3 3397 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ 𝑌 → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ 𝑥𝑆(𝐹𝑧)))
8433, 83syl 17 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ 𝑥𝑆(𝐹𝑧)))
85 isorel 6616 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ ((𝐹𝑥) ∈ 𝑋𝑧𝑋)) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8638, 43, 30, 85syl12anc 1364 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8748breq1d 4695 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹‘(𝐹𝑥))𝑆(𝐹𝑧) ↔ 𝑥𝑆(𝐹𝑧)))
8886, 87bitrd 268 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧𝑥𝑆(𝐹𝑧)))
8988notbid 307 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ (𝐹𝑥)𝑅𝑧 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
9084, 89bitr4d 271 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ (𝐹𝑥)𝑅𝑧))
9179, 80, 903bitr2d 296 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ ¬ (𝐹𝑥)𝑅𝑧))
9291rabbi2dva 3854 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
9377, 92eqtr3d 2687 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
947ordtopn2 21047 . . . . . . . 8 ((𝑅𝑉 ∧ (𝐹𝑥) ∈ 𝑋) → {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
9556, 42, 94syl2anc 694 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
9693, 95eqeltrd 2730 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
9796ralrimiva 2995 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
98 rabexg 4844 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
9964, 98syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
10099ralrimivw 2996 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
101 eqid 2651 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
102 imaeq2 5497 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
103102eleq1d 2715 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
104101, 103ralrnmpt 6408 . . . . . 6 (∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V → (∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
105100, 104syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅) ↔ ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
10697, 105mpbird 247 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅))
107 ralunb 3827 . . . 4 (∀𝑧 ∈ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))(𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅) ∧ ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅)))
10873, 106, 107sylanbrc 699 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))(𝐹𝑧) ∈ (ordTop‘𝑅))
109 ralunb 3827 . . 3 (∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))(𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (∀𝑧 ∈ {𝑌} (𝐹𝑧) ∈ (ordTop‘𝑅) ∧ ∀𝑧 ∈ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))(𝐹𝑧) ∈ (ordTop‘𝑅)))
11017, 108, 109sylanbrc 699 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))(𝐹𝑧) ∈ (ordTop‘𝑅))
111 eqid 2651 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
112 eqid 2651 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
11361, 111, 112ordtuni 21042 . . . . . 6 (𝑆𝑊𝑌 = ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))
114113, 63eqeltrrd 2731 . . . . 5 (𝑆𝑊 ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
115 uniexb 7015 . . . . 5 (({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V ↔ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
116114, 115sylibr 224 . . . 4 (𝑆𝑊 → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
1171163ad2ant2 1103 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
11861, 111, 112ordtval 21041 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
1191183ad2ant2 1103 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
12061ordttopon 21045 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1211203ad2ant2 1103 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1229, 117, 119, 121subbascn 21106 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))(𝐹𝑧) ∈ (ordTop‘𝑅))))
1234, 110, 122mpbir2and 977 1 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cun 3605  cin 3606  wss 3607  {csn 4210   cuni 4468   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  ficfi 8357  topGenctg 16145  ordTopcordt 16206  TopOnctopon 20763   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-topgen 16151  df-ordt 16208  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079
This theorem is referenced by:  ordthmeo  21653  xrmulc1cn  30104
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