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Theorem ordthmeolem 21884
Description: Lemma for ordthmeo 21885. (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 6765 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) → 𝐹:𝑋1-1-onto𝑌)
213ad2ant3 1165 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3 f1of 6320 . . 3 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
42, 3syl 17 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋𝑌)
5 fimacnv 6537 . . . . . . 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 21277 . . . . . . . 8 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
983ad2ant1 1163 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
10 toponmax 21010 . . . . . . 7 ((ordTop‘𝑅) ∈ (TopOn‘𝑋) → 𝑋 ∈ (ordTop‘𝑅))
119, 10syl 17 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑋 ∈ (ordTop‘𝑅))
126, 11eqeltrd 2844 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹𝑌) ∈ (ordTop‘𝑅))
13 elsni 4351 . . . . . . 7 (𝑧 ∈ {𝑌} → 𝑧 = 𝑌)
1413imaeq2d 5648 . . . . . 6 (𝑧 ∈ {𝑌} → (𝐹𝑧) = (𝐹𝑌))
1514eleq1d 2829 . . . . 5 (𝑧 ∈ {𝑌} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹𝑌) ∈ (ordTop‘𝑅)))
1612, 15syl5ibrcom 238 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝑧 ∈ {𝑌} → (𝐹𝑧) ∈ (ordTop‘𝑅)))
1716ralrimiv 3112 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ {𝑌} (𝐹𝑧) ∈ (ordTop‘𝑅))
18 cnvimass 5667 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ dom 𝐹
19 f1odm 6324 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌 → dom 𝐹 = 𝑋)
202, 19syl 17 . . . . . . . . . . 11 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → dom 𝐹 = 𝑋)
2120adantr 472 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → dom 𝐹 = 𝑋)
2218, 21syl5sseq 3813 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋)
23 sseqin2 3979 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
2422, 23sylib 209 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
252ad2antrr 717 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹:𝑋1-1-onto𝑌)
26 f1ofn 6321 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌𝐹 Fn 𝑋)
2725, 26syl 17 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Fn 𝑋)
28 elpreima 6527 . . . . . . . . . . 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 477 . . . . . . . . . . 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 472 . . . . . . . . . . . . 13 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝐹:𝑋𝑌)
3332ffvelrnda 6549 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ 𝑌)
34 breq1 4812 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑦𝑆𝑥 ↔ (𝐹𝑧)𝑆𝑥))
3534notbid 309 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑦𝑆𝑥 ↔ ¬ (𝐹𝑧)𝑆𝑥))
3635elrab3 3521 . . . . . . . . . . . 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 1273 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌))
39 f1ocnv 6332 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
40 f1of 6320 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
412, 39, 403syl 18 . . . . . . . . . . . . . . . 16 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑌𝑋)
4241ffvelrnda 6549 . . . . . . . . . . . . . . 15 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
4342adantr 472 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑥) ∈ 𝑋)
44 isorel 6768 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ (𝑧𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
4538, 30, 43, 44syl12anc 865 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
46 f1ocnvfv2 6725 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
472, 46sylan 575 . . . . . . . . . . . . . . 15 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
4847adantr 472 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹‘(𝐹𝑥)) = 𝑥)
4948breq2d 4821 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧)𝑆(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑧)𝑆𝑥))
5045, 49bitrd 270 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆𝑥))
5150notbid 309 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ 𝑧𝑅(𝐹𝑥) ↔ ¬ (𝐹𝑧)𝑆𝑥))
5237, 51bitr4d 273 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5329, 31, 523bitr2d 298 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5453rabbi2dva 3981 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
5524, 54eqtr3d 2801 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
56 simpl1 1242 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝑅𝑉)
577ordtopn1 21278 . . . . . . . 8 ((𝑅𝑉 ∧ (𝐹𝑥) ∈ 𝑋) → {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)} ∈ (ordTop‘𝑅))
5856, 42, 57syl2anc 579 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)} ∈ (ordTop‘𝑅))
5955, 58eqeltrd 2844 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
6059ralrimiva 3113 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
61 ordthmeo.2 . . . . . . . . . 10 𝑌 = dom 𝑆
62 dmexg 7295 . . . . . . . . . 10 (𝑆𝑊 → dom 𝑆 ∈ V)
6361, 62syl5eqel 2848 . . . . . . . . 9 (𝑆𝑊𝑌 ∈ V)
64633ad2ant2 1164 . . . . . . . 8 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑌 ∈ V)
65 rabexg 4972 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6664, 65syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6766ralrimivw 3114 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
68 eqid 2765 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
69 imaeq2 5644 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
7069eleq1d 2829 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
7168, 70ralrnmpt 6558 . . . . . 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 248 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅))
74 cnvimass 5667 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ dom 𝐹
7574, 21syl5sseq 3813 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋)
76 sseqin2 3979 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
7775, 76sylib 209 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
78 elpreima 6527 . . . . . . . . . . 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 4813 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑥𝑆𝑦𝑥𝑆(𝐹𝑧)))
8281notbid 309 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
8382elrab3 3521 . . . . . . . . . . . 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 6768 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ ((𝐹𝑥) ∈ 𝑋𝑧𝑋)) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8638, 43, 30, 85syl12anc 865 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8748breq1d 4819 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹‘(𝐹𝑥))𝑆(𝐹𝑧) ↔ 𝑥𝑆(𝐹𝑧)))
8886, 87bitrd 270 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧𝑥𝑆(𝐹𝑧)))
8988notbid 309 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ (𝐹𝑥)𝑅𝑧 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
9084, 89bitr4d 273 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ (𝐹𝑥)𝑅𝑧))
9179, 80, 903bitr2d 298 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ ¬ (𝐹𝑥)𝑅𝑧))
9291rabbi2dva 3981 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
9377, 92eqtr3d 2801 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
947ordtopn2 21279 . . . . . . . 8 ((𝑅𝑉 ∧ (𝐹𝑥) ∈ 𝑋) → {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
9556, 42, 94syl2anc 579 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧} ∈ (ordTop‘𝑅))
9693, 95eqeltrd 2844 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
9796ralrimiva 3113 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
98 rabexg 4972 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
9964, 98syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
10099ralrimivw 3114 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
101 eqid 2765 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
102 imaeq2 5644 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
103102eleq1d 2829 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
104101, 103ralrnmpt 6558 . . . . . 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 248 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅))
107 ralunb 3956 . . . 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 578 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))(𝐹𝑧) ∈ (ordTop‘𝑅))
109 ralunb 3956 . . 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 578 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})))(𝐹𝑧) ∈ (ordTop‘𝑅))
111 eqid 2765 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
112 eqid 2765 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
11361, 111, 112ordtuni 21274 . . . . . 6 (𝑆𝑊𝑌 = ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))
114113, 63eqeltrrd 2845 . . . . 5 (𝑆𝑊 ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
115 uniexb 7171 . . . . 5 (({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V ↔ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
116114, 115sylibr 225 . . . 4 (𝑆𝑊 → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
1171163ad2ant2 1164 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
11861, 111, 112ordtval 21273 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
1191183ad2ant2 1164 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
12061ordttopon 21277 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1211203ad2ant2 1164 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1229, 117, 119, 121subbascn 21338 . 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 704 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cun 3730  cin 3731  wss 3732  {csn 4334   cuni 4594   class class class wbr 4809  cmpt 4888  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068   Isom wiso 6069  (class class class)co 6842  ficfi 8523  topGenctg 16364  ordTopcordt 16425  TopOnctopon 20994   Cn ccn 21308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-topgen 16370  df-ordt 16427  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311
This theorem is referenced by:  ordthmeo  21885  xrmulc1cn  30423
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