MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtbaslem Structured version   Visualization version   GIF version

Theorem ordtbaslem 21040
Description: Lemma for ordtbas 21044. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
ordtval.1 𝑋 = dom 𝑅
ordtval.2 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
Assertion
Ref Expression
ordtbaslem (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem ordtbaslem
Dummy variables 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3anrot 1060 . . . . . . . . . . . . 13 ((𝑦𝑋𝑎𝑋𝑏𝑋) ↔ (𝑎𝑋𝑏𝑋𝑦𝑋))
2 ordtval.1 . . . . . . . . . . . . . 14 𝑋 = dom 𝑅
32tsrlemax 17267 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑦𝑋𝑎𝑋𝑏𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎𝑦𝑅𝑏)))
41, 3sylan2br 492 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋𝑦𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎𝑦𝑅𝑏)))
543exp2 1307 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → (𝑎𝑋 → (𝑏𝑋 → (𝑦𝑋 → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎𝑦𝑅𝑏))))))
65imp42 619 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑦𝑋) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎𝑦𝑅𝑏)))
76notbid 307 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑦𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ ¬ (𝑦𝑅𝑎𝑦𝑅𝑏)))
8 ioran 510 . . . . . . . . 9 (¬ (𝑦𝑅𝑎𝑦𝑅𝑏) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏))
97, 8syl6bb 276 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑦𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)))
109rabbidva 3219 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
11 ifcl 4163 . . . . . . . . . 10 ((𝑏𝑋𝑎𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
1211ancoms 468 . . . . . . . . 9 ((𝑎𝑋𝑏𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
1312adantl 481 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
14 dmexg 7139 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
152, 14syl5eqel 2734 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
1615adantr 480 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑋 ∈ V)
17 rabexg 4844 . . . . . . . . . 10 (𝑋 ∈ V → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
1816, 17syl 17 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
1910, 18eqeltrd 2730 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V)
20 eqid 2651 . . . . . . . . . 10 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
21 breq2 4689 . . . . . . . . . . . 12 (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (𝑦𝑅𝑥𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
2221notbid 307 . . . . . . . . . . 11 (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
2322rabbidv 3220 . . . . . . . . . 10 (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)})
2420, 23elrnmpt1s 5405 . . . . . . . . 9 ((if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V) → {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
25 ordtval.2 . . . . . . . . 9 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
2624, 25syl6eleqr 2741 . . . . . . . 8 ((if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V) → {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
2713, 19, 26syl2anc 694 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
2810, 27eqeltrrd 2731 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
2928ralrimivva 3000 . . . . 5 (𝑅 ∈ TosetRel → ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
30 rabexg 4844 . . . . . . . 8 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
3115, 30syl 17 . . . . . . 7 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
3231ralrimivw 2996 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑎𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
33 breq2 4689 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑦𝑅𝑥𝑦𝑅𝑎))
3433notbid 307 . . . . . . . . 9 (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
3534rabbidv 3220 . . . . . . . 8 (𝑥 = 𝑎 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
3635cbvmptv 4783 . . . . . . 7 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
37 ineq1 3840 . . . . . . . . . 10 (𝑧 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) = ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}))
38 inrab 3932 . . . . . . . . . 10 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)}
3937, 38syl6eq 2701 . . . . . . . . 9 (𝑧 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
4039eleq1d 2715 . . . . . . . 8 (𝑧 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} → ((𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4140ralbidv 3015 . . . . . . 7 (𝑧 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} → (∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4236, 41ralrnmpt 6408 . . . . . 6 (∀𝑎𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → (∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4332, 42syl 17 . . . . 5 (𝑅 ∈ TosetRel → (∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4429, 43mpbird 247 . . . 4 (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴)
45 rabexg 4844 . . . . . . . 8 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
4615, 45syl 17 . . . . . . 7 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
4746ralrimivw 2996 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑏𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
48 breq2 4689 . . . . . . . . . 10 (𝑥 = 𝑏 → (𝑦𝑅𝑥𝑦𝑅𝑏))
4948notbid 307 . . . . . . . . 9 (𝑥 = 𝑏 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑏))
5049rabbidv 3220 . . . . . . . 8 (𝑥 = 𝑏 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏})
5150cbvmptv 4783 . . . . . . 7 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑏𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏})
52 ineq2 3841 . . . . . . . 8 (𝑤 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏} → (𝑧𝑤) = (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}))
5352eleq1d 2715 . . . . . . 7 (𝑤 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏} → ((𝑧𝑤) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5451, 53ralrnmpt 6408 . . . . . 6 (∀𝑏𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V → (∀𝑤 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧𝑤) ∈ 𝐴 ↔ ∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5547, 54syl 17 . . . . 5 (𝑅 ∈ TosetRel → (∀𝑤 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧𝑤) ∈ 𝐴 ↔ ∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5655ralbidv 3015 . . . 4 (𝑅 ∈ TosetRel → (∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧𝑤) ∈ 𝐴 ↔ ∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏𝑋 (𝑧 ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5744, 56mpbird 247 . . 3 (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧𝑤) ∈ 𝐴)
5825raleqi 3172 . . . 4 (∀𝑤𝐴 (𝑧𝑤) ∈ 𝐴 ↔ ∀𝑤 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧𝑤) ∈ 𝐴)
5925, 58raleqbii 3019 . . 3 (∀𝑧𝐴𝑤𝐴 (𝑧𝑤) ∈ 𝐴 ↔ ∀𝑧 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧𝑤) ∈ 𝐴)
6057, 59sylibr 224 . 2 (𝑅 ∈ TosetRel → ∀𝑧𝐴𝑤𝐴 (𝑧𝑤) ∈ 𝐴)
61 pwexg 4880 . . . . 5 (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
6215, 61syl 17 . . . 4 (𝑅 ∈ TosetRel → 𝒫 𝑋 ∈ V)
63 ssrab2 3720 . . . . . . . 8 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
6415adantr 480 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → 𝑋 ∈ V)
65 elpw2g 4857 . . . . . . . . 9 (𝑋 ∈ V → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
6664, 65syl 17 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
6763, 66mpbiri 248 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
6867, 20fmptd 6425 . . . . . 6 (𝑅 ∈ TosetRel → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
69 frn 6091 . . . . . 6 ((𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
7068, 69syl 17 . . . . 5 (𝑅 ∈ TosetRel → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
7125, 70syl5eqss 3682 . . . 4 (𝑅 ∈ TosetRel → 𝐴 ⊆ 𝒫 𝑋)
7262, 71ssexd 4838 . . 3 (𝑅 ∈ TosetRel → 𝐴 ∈ V)
73 inficl 8372 . . 3 (𝐴 ∈ V → (∀𝑧𝐴𝑤𝐴 (𝑧𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
7472, 73syl 17 . 2 (𝑅 ∈ TosetRel → (∀𝑧𝐴𝑤𝐴 (𝑧𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
7560, 74mpbid 222 1 (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cin 3606  wss 3607  ifcif 4119  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  wf 5922  cfv 5926  ficfi 8357   TosetRel ctsr 17246
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-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-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-ps 17247  df-tsr 17248
This theorem is referenced by:  ordtbas2  21043
  Copyright terms: Public domain W3C validator