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Theorem ordthmeolem 22325
Description: Lemma for ordthmeo 22326. (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 7071 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) → 𝐹:𝑋1-1-onto𝑌)
213ad2ant3 1129 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3 f1of 6611 . . 3 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
42, 3syl 17 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋𝑌)
5 fimacnv 6834 . . . . . . 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 21717 . . . . . . . 8 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
983ad2ant1 1127 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
10 toponmax 21450 . . . . . . 7 ((ordTop‘𝑅) ∈ (TopOn‘𝑋) → 𝑋 ∈ (ordTop‘𝑅))
119, 10syl 17 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑋 ∈ (ordTop‘𝑅))
126, 11eqeltrd 2917 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹𝑌) ∈ (ordTop‘𝑅))
13 elsni 4580 . . . . . . 7 (𝑧 ∈ {𝑌} → 𝑧 = 𝑌)
1413imaeq2d 5926 . . . . . 6 (𝑧 ∈ {𝑌} → (𝐹𝑧) = (𝐹𝑌))
1514eleq1d 2901 . . . . 5 (𝑧 ∈ {𝑌} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹𝑌) ∈ (ordTop‘𝑅)))
1612, 15syl5ibrcom 248 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝑧 ∈ {𝑌} → (𝐹𝑧) ∈ (ordTop‘𝑅)))
1716ralrimiv 3185 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ {𝑌} (𝐹𝑧) ∈ (ordTop‘𝑅))
18 cnvimass 5946 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ dom 𝐹
19 f1odm 6615 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌 → dom 𝐹 = 𝑋)
202, 19syl 17 . . . . . . . . . . 11 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → dom 𝐹 = 𝑋)
2120adantr 481 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → dom 𝐹 = 𝑋)
2218, 21sseqtrid 4022 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋)
23 sseqin2 4195 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
2422, 23sylib 219 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
252ad2antrr 722 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹:𝑋1-1-onto𝑌)
26 f1ofn 6612 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌𝐹 Fn 𝑋)
2725, 26syl 17 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Fn 𝑋)
28 elpreima 6823 . . . . . . . . . . 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 485 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝑧𝑋)
3130biantrurd 533 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})))
324adantr 481 . . . . . . . . . . . . 13 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝐹:𝑋𝑌)
3332ffvelrnda 6846 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ 𝑌)
34 breq1 5065 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑦𝑆𝑥 ↔ (𝐹𝑧)𝑆𝑥))
3534notbid 319 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑦𝑆𝑥 ↔ ¬ (𝐹𝑧)𝑆𝑥))
3635elrab3 3684 . . . . . . . . . . . 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 1208 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌))
39 f1ocnv 6623 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
40 f1of 6611 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
412, 39, 403syl 18 . . . . . . . . . . . . . . . 16 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑌𝑋)
4241ffvelrnda 6846 . . . . . . . . . . . . . . 15 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
4342adantr 481 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑥) ∈ 𝑋)
44 isorel 7074 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ (𝑧𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
4538, 30, 43, 44syl12anc 834 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
46 f1ocnvfv2 7031 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
472, 46sylan 580 . . . . . . . . . . . . . . 15 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
4847adantr 481 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹‘(𝐹𝑥)) = 𝑥)
4948breq2d 5074 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧)𝑆(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑧)𝑆𝑥))
5045, 49bitrd 280 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆𝑥))
5150notbid 319 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ 𝑧𝑅(𝐹𝑥) ↔ ¬ (𝐹𝑧)𝑆𝑥))
5237, 51bitr4d 283 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5329, 31, 523bitr2d 308 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5453rabbi2dva 4197 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
5524, 54eqtr3d 2862 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
56 simpl1 1185 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝑅𝑉)
577ordtopn1 21718 . . . . . . . 8 ((𝑅𝑉 ∧ (𝐹𝑥) ∈ 𝑋) → {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)} ∈ (ordTop‘𝑅))
5856, 42, 57syl2anc 584 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)} ∈ (ordTop‘𝑅))
5955, 58eqeltrd 2917 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
6059ralrimiva 3186 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
61 ordthmeo.2 . . . . . . . . . 10 𝑌 = dom 𝑆
62 dmexg 7604 . . . . . . . . . 10 (𝑆𝑊 → dom 𝑆 ∈ V)
6361, 62eqeltrid 2921 . . . . . . . . 9 (𝑆𝑊𝑌 ∈ V)
64633ad2ant2 1128 . . . . . . . 8 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑌 ∈ V)
65 rabexg 5230 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6664, 65syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6766ralrimivw 3187 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
68 eqid 2825 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
69 imaeq2 5922 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
7069eleq1d 2901 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
7168, 70ralrnmptw 6855 . . . . . 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 258 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅))
74 cnvimass 5946 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ dom 𝐹
7574, 21sseqtrid 4022 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋)
76 sseqin2 4195 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
7775, 76sylib 219 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
78 elpreima 6823 . . . . . . . . . . 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 533 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))
81 breq2 5066 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑥𝑆𝑦𝑥𝑆(𝐹𝑧)))
8281notbid 319 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
8382elrab3 3684 . . . . . . . . . . . 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 7074 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ ((𝐹𝑥) ∈ 𝑋𝑧𝑋)) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8638, 43, 30, 85syl12anc 834 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8748breq1d 5072 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹‘(𝐹𝑥))𝑆(𝐹𝑧) ↔ 𝑥𝑆(𝐹𝑧)))
8886, 87bitrd 280 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧𝑥𝑆(𝐹𝑧)))
8988notbid 319 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ (𝐹𝑥)𝑅𝑧 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
9084, 89bitr4d 283 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ (𝐹𝑥)𝑅𝑧))
9179, 80, 903bitr2d 308 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ ¬ (𝐹𝑥)𝑅𝑧))
9291rabbi2dva 4197 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
9377, 92eqtr3d 2862 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
947ordtopn2 21719 . . . . . . . 8 ((𝑅𝑉 ∧ (𝐹𝑥) ∈ 𝑋) → {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
9556, 42, 94syl2anc 584 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
9693, 95eqeltrd 2917 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
9796ralrimiva 3186 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
98 rabexg 5230 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
9964, 98syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
10099ralrimivw 3187 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
101 eqid 2825 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
102 imaeq2 5922 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
103102eleq1d 2901 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
104101, 103ralrnmptw 6855 . . . . . 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 258 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅))
107 ralunb 4170 . . . 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 583 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))(𝐹𝑧) ∈ (ordTop‘𝑅))
109 ralunb 4170 . . 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 583 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))(𝐹𝑧) ∈ (ordTop‘𝑅))
111 eqid 2825 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
112 eqid 2825 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
11361, 111, 112ordtuni 21714 . . . . . 6 (𝑆𝑊𝑌 = ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))
114113, 63eqeltrrd 2918 . . . . 5 (𝑆𝑊 ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
115 uniexb 7478 . . . . 5 (({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V ↔ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
116114, 115sylibr 235 . . . 4 (𝑆𝑊 → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
1171163ad2ant2 1128 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
11861, 111, 112ordtval 21713 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
1191183ad2ant2 1128 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
12061ordttopon 21717 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1211203ad2ant2 1128 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1229, 117, 119, 121subbascn 21778 . 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 709 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3142  {crab 3146  Vcvv 3499  cun 3937  cin 3938  wss 3939  {csn 4563   cuni 4836   class class class wbr 5062  cmpt 5142  ccnv 5552  dom cdm 5553  ran crn 5554  cima 5556   Fn wfn 6346  wf 6347  1-1-ontowf1o 6350  cfv 6351   Isom wiso 6352  (class class class)co 7151  ficfi 8866  topGenctg 16703  ordTopcordt 16764  TopOnctopon 21434   Cn ccn 21748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-fin 8505  df-fi 8867  df-topgen 16709  df-ordt 16766  df-top 21418  df-topon 21435  df-bases 21470  df-cn 21751
This theorem is referenced by:  ordthmeo  22326  xrmulc1cn  31059
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