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Theorem ordthmeolem 23152
Description: Lemma for ordthmeo 23153. (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 7268 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) → 𝐹:𝑋1-1-onto𝑌)
213ad2ant3 1135 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3 f1of 6784 . . 3 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
42, 3syl 17 . 2 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑋𝑌)
5 fimacnv 6690 . . . . . . 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 22544 . . . . . . . 8 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
983ad2ant1 1133 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
10 toponmax 22275 . . . . . . 7 ((ordTop‘𝑅) ∈ (TopOn‘𝑋) → 𝑋 ∈ (ordTop‘𝑅))
119, 10syl 17 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑋 ∈ (ordTop‘𝑅))
126, 11eqeltrd 2838 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝐹𝑌) ∈ (ordTop‘𝑅))
13 elsni 4603 . . . . . . 7 (𝑧 ∈ {𝑌} → 𝑧 = 𝑌)
1413imaeq2d 6013 . . . . . 6 (𝑧 ∈ {𝑌} → (𝐹𝑧) = (𝐹𝑌))
1514eleq1d 2822 . . . . 5 (𝑧 ∈ {𝑌} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹𝑌) ∈ (ordTop‘𝑅)))
1612, 15syl5ibrcom 246 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (𝑧 ∈ {𝑌} → (𝐹𝑧) ∈ (ordTop‘𝑅)))
1716ralrimiv 3142 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ {𝑌} (𝐹𝑧) ∈ (ordTop‘𝑅))
18 cnvimass 6033 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ dom 𝐹
19 f1odm 6788 . . . . . . . . . . . 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 3996 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋)
23 sseqin2 4175 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
2422, 23sylib 217 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
252ad2antrr 724 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹:𝑋1-1-onto𝑌)
26 f1ofn 6785 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌𝐹 Fn 𝑋)
2725, 26syl 17 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Fn 𝑋)
28 elpreima 7008 . . . . . . . . . . 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)
3332ffvelcdmda 7035 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ 𝑌)
34 breq1 5108 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑦𝑆𝑥 ↔ (𝐹𝑧)𝑆𝑥))
3534notbid 317 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑦𝑆𝑥 ↔ ¬ (𝐹𝑧)𝑆𝑥))
3635elrab3 3646 . . . . . . . . . . . 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 1214 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌))
39 f1ocnv 6796 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
40 f1of 6784 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
412, 39, 403syl 18 . . . . . . . . . . . . . . . 16 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹:𝑌𝑋)
4241ffvelcdmda 7035 . . . . . . . . . . . . . . 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 7271 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ (𝑧𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
4538, 30, 43, 44syl12anc 835 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆(𝐹‘(𝐹𝑥))))
46 f1ocnvfv2 7223 . . . . . . . . . . . . . . . 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 5117 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧)𝑆(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑧)𝑆𝑥))
5045, 49bitrd 278 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧𝑅(𝐹𝑥) ↔ (𝐹𝑧)𝑆𝑥))
5150notbid 317 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ 𝑧𝑅(𝐹𝑥) ↔ ¬ (𝐹𝑧)𝑆𝑥))
5237, 51bitr4d 281 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5329, 31, 523bitr2d 306 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ↔ ¬ 𝑧𝑅(𝐹𝑥)))
5453rabbi2dva 4177 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
5524, 54eqtr3d 2778 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = {𝑧𝑋 ∣ ¬ 𝑧𝑅(𝐹𝑥)})
56 simpl1 1191 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → 𝑅𝑉)
577ordtopn1 22545 . . . . . . . 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 2838 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
6059ralrimiva 3143 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅))
61 ordthmeo.2 . . . . . . . . . 10 𝑌 = dom 𝑆
62 dmexg 7840 . . . . . . . . . 10 (𝑆𝑊 → dom 𝑆 ∈ V)
6361, 62eqeltrid 2842 . . . . . . . . 9 (𝑆𝑊𝑌 ∈ V)
64633ad2ant2 1134 . . . . . . . 8 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝑌 ∈ V)
65 rabexg 5288 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6664, 65syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
6766ralrimivw 3147 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} ∈ V)
68 eqid 2736 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
69 imaeq2 6009 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}))
7069eleq1d 2822 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∈ (ordTop‘𝑅)))
7168, 70ralrnmptw 7044 . . . . . 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 256 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})(𝐹𝑧) ∈ (ordTop‘𝑅))
74 cnvimass 6033 . . . . . . . . . 10 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ dom 𝐹
7574, 21sseqtrid 3996 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋)
76 sseqin2 4175 . . . . . . . . 9 ((𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ⊆ 𝑋 ↔ (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
7775, 76sylib 217 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
78 elpreima 7008 . . . . . . . . . . 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 5109 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑧) → (𝑥𝑆𝑦𝑥𝑆(𝐹𝑧)))
8281notbid 317 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑧) → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
8382elrab3 3646 . . . . . . . . . . . 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 7271 . . . . . . . . . . . . . 14 ((𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌) ∧ ((𝐹𝑥) ∈ 𝑋𝑧𝑋)) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8638, 43, 30, 85syl12anc 835 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧 ↔ (𝐹‘(𝐹𝑥))𝑆(𝐹𝑧)))
8748breq1d 5115 . . . . . . . . . . . . 13 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹‘(𝐹𝑥))𝑆(𝐹𝑧) ↔ 𝑥𝑆(𝐹𝑧)))
8886, 87bitrd 278 . . . . . . . . . . . 12 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑥)𝑅𝑧𝑥𝑆(𝐹𝑧)))
8988notbid 317 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (¬ (𝐹𝑥)𝑅𝑧 ↔ ¬ 𝑥𝑆(𝐹𝑧)))
9084, 89bitr4d 281 . . . . . . . . . 10 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ↔ ¬ (𝐹𝑥)𝑅𝑧))
9179, 80, 903bitr2d 306 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) ∧ 𝑧𝑋) → (𝑧 ∈ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ↔ ¬ (𝐹𝑥)𝑅𝑧))
9291rabbi2dva 4177 . . . . . . . 8 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝑋 ∩ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
9377, 92eqtr3d 2778 . . . . . . 7 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = {𝑧𝑋 ∣ ¬ (𝐹𝑥)𝑅𝑧})
947ordtopn2 22546 . . . . . . . 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 2838 . . . . . 6 (((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) ∧ 𝑥𝑌) → (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
9796ralrimiva 3143 . . . . 5 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅))
98 rabexg 5288 . . . . . . . 8 (𝑌 ∈ V → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
9964, 98syl 17 . . . . . . 7 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
10099ralrimivw 3147 . . . . . 6 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑥𝑌 {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} ∈ V)
101 eqid 2736 . . . . . . 7 (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
102 imaeq2 6009 . . . . . . . 8 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → (𝐹𝑧) = (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))
103102eleq1d 2822 . . . . . . 7 (𝑧 = {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦} → ((𝐹𝑧) ∈ (ordTop‘𝑅) ↔ (𝐹 “ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) ∈ (ordTop‘𝑅)))
104101, 103ralrnmptw 7044 . . . . . 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 256 . . . 4 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ∀𝑧 ∈ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})(𝐹𝑧) ∈ (ordTop‘𝑅))
107 ralunb 4151 . . . 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 4151 . . 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 2736 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥})
112 eqid 2736 . . . . . . 7 ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}) = ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦})
11361, 111, 112ordtuni 22541 . . . . . 6 (𝑆𝑊𝑌 = ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))
114113, 63eqeltrrd 2839 . . . . 5 (𝑆𝑊 ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
115 uniexb 7698 . . . . 5 (({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V ↔ ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
116114, 115sylibr 233 . . . 4 (𝑆𝑊 → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
1171163ad2ant2 1134 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → ({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))) ∈ V)
11861, 111, 112ordtval 22540 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
1191183ad2ant2 1134 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) = (topGen‘(fi‘({𝑌} ∪ (ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑦𝑆𝑥}) ∪ ran (𝑥𝑌 ↦ {𝑦𝑌 ∣ ¬ 𝑥𝑆𝑦}))))))
12061ordttopon 22544 . . . 4 (𝑆𝑊 → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1211203ad2ant2 1134 . . 3 ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → (ordTop‘𝑆) ∈ (TopOn‘𝑌))
1229, 117, 119, 121subbascn 22605 . 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 711 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cun 3908  cin 3909  wss 3910  {csn 4586   cuni 4865   class class class wbr 5105  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496   Isom wiso 6497  (class class class)co 7357  ficfi 9346  topGenctg 17319  ordTopcordt 17381  TopOnctopon 22259   Cn ccn 22575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-topgen 17325  df-ordt 17383  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578
This theorem is referenced by:  ordthmeo  23153  xrmulc1cn  32511
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