ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr1onlembfn GIF version

Theorem tfr1onlembfn 6013
Description: Lemma for tfr1on 6019. The union of 𝐵 is a function defined on 𝑥. (Contributed by Jim Kingdon, 15-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlembacc.3 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
tfr1onlembacc.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfr1onlembacc.4 (𝜑𝐷𝑋)
tfr1onlembacc.5 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
Assertion
Ref Expression
tfr1onlembfn (𝜑 𝐵 Fn 𝐷)
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑧   𝐷,𝑓,𝑔,𝑥   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,,𝑥,𝑧   𝑦,𝑔,𝑧   𝐵,𝑔,,𝑧,𝑤   𝐷,,𝑧   ,𝐺,𝑧,𝑤,𝑓,𝑦,𝑥   𝑔,𝑋,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑤)   𝐴(𝑦,𝑤)   𝐵(𝑥,𝑦,𝑓)   𝐷(𝑦,𝑤)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,)

Proof of Theorem tfr1onlembfn
StepHypRef Expression
1 tfr1on.f . . . . . 6 𝐹 = recs(𝐺)
2 tfr1on.g . . . . . 6 (𝜑 → Fun 𝐺)
3 tfr1on.x . . . . . 6 (𝜑 → Ord 𝑋)
4 tfr1on.ex . . . . . 6 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
5 tfr1onlemsucfn.1 . . . . . 6 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6 tfr1onlembacc.3 . . . . . 6 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7 tfr1onlembacc.u . . . . . 6 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfr1onlembacc.4 . . . . . 6 (𝜑𝐷𝑋)
9 tfr1onlembacc.5 . . . . . 6 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
101, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembacc 6011 . . . . 5 (𝜑𝐵𝐴)
1110unissd 3645 . . . 4 (𝜑 𝐵 𝐴)
125, 3tfr1onlemssrecs 6008 . . . 4 (𝜑 𝐴 ⊆ recs(𝐺))
1311, 12sstrd 3018 . . 3 (𝜑 𝐵 ⊆ recs(𝐺))
14 tfrfun 5989 . . 3 Fun recs(𝐺)
15 funss 4970 . . 3 ( 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun 𝐵))
1613, 14, 15mpisyl 1376 . 2 (𝜑 → Fun 𝐵)
17 simpr3 947 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
18 simpl 107 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝜑)
193adantr 270 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → Ord 𝑋)
20 simpr 108 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝑧𝐷)
218adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝐷𝑋)
2220, 21jca 300 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → (𝑧𝐷𝐷𝑋))
23 ordtr1 4171 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑋 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
2419, 22, 23sylc 61 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝑧𝑋)
2518, 24jca 300 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐷) → (𝜑𝑧𝑋))
262ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Fun 𝐺)
273ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝑋)
2843adant1r 1163 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑋) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
29283adant1r 1163 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
30 simplr 497 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
31 simpr1 945 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
32 simpr2 946 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
331, 26, 27, 29, 5, 30, 31, 32tfr1onlemsucfn 6009 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
3425, 33sylan 277 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
35 dffn2 5098 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V)
3634, 35sylib 120 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V)
37 fssxp 5109 . . . . . . . . . . . . . . 15 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (suc 𝑧 × V))
3836, 37syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (suc 𝑧 × V))
39 ordelon 4166 . . . . . . . . . . . . . . . . . . 19 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
403, 8, 39syl2anc 403 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 ∈ On)
41 eloni 4158 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ On → Ord 𝐷)
4240, 41syl 14 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord 𝐷)
4342ad2antrr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝐷)
44 simplr 497 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝐷)
45 ordsucss 4276 . . . . . . . . . . . . . . . 16 (Ord 𝐷 → (𝑧𝐷 → suc 𝑧𝐷))
4643, 44, 45sylc 61 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → suc 𝑧𝐷)
47 xpss1 4496 . . . . . . . . . . . . . . 15 (suc 𝑧𝐷 → (suc 𝑧 × V) ⊆ (𝐷 × V))
4846, 47syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (suc 𝑧 × V) ⊆ (𝐷 × V))
4938, 48sstrd 3018 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × V))
50 vex 2613 . . . . . . . . . . . . . . 15 𝑔 ∈ V
51 vex 2613 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
5218adantr 270 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝜑)
5324adantr 270 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
54 simpr1 945 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
55 fneq2 5039 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
5655imbi1d 229 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5756albidv 1747 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5843expia 1141 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
5958alrimiv 1797 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6059ralrimiva 2439 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
61603ad2ant1 960 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
62 simp2 940 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑧𝑋)
6357, 61, 62rspcdva 2715 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
64 simp3 941 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑔 Fn 𝑧)
65 fneq1 5038 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
66 fveq2 5229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
6766eleq1d 2151 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
6865, 67imbi12d 232 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
6968spv 1783 . . . . . . . . . . . . . . . . . . 19 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
7063, 64, 69sylc 61 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → (𝐺𝑔) ∈ V)
7152, 53, 54, 70syl3anc 1170 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝐺𝑔) ∈ V)
72 opexg 4011 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
7351, 71, 72sylancr 405 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
74 snexg 3976 . . . . . . . . . . . . . . . 16 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
7573, 74syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
76 unexg 4224 . . . . . . . . . . . . . . 15 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
7750, 75, 76sylancr 405 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
78 elpwg 3408 . . . . . . . . . . . . . 14 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × V)))
7977, 78syl 14 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × V)))
8049, 79mpbird 165 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × V))
8117, 80eqeltrd 2159 . . . . . . . . . . 11 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ∈ 𝒫 (𝐷 × V))
8281ex 113 . . . . . . . . . 10 ((𝜑𝑧𝐷) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8382exlimdv 1742 . . . . . . . . 9 ((𝜑𝑧𝐷) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8483rexlimdva 2482 . . . . . . . 8 (𝜑 → (∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8584abssdv 3077 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝒫 (𝐷 × V))
866, 85syl5eqss 3052 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝐷 × V))
87 sspwuni 3780 . . . . . 6 (𝐵 ⊆ 𝒫 (𝐷 × V) ↔ 𝐵 ⊆ (𝐷 × V))
8886, 87sylib 120 . . . . 5 (𝜑 𝐵 ⊆ (𝐷 × V))
89 dmss 4582 . . . . 5 ( 𝐵 ⊆ (𝐷 × V) → dom 𝐵 ⊆ dom (𝐷 × V))
9088, 89syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝐷 × V))
91 dmxpss 4803 . . . 4 dom (𝐷 × V) ⊆ 𝐷
9290, 91syl6ss 3020 . . 3 (𝜑 → dom 𝐵𝐷)
931, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembxssdm 6012 . . 3 (𝜑𝐷 ⊆ dom 𝐵)
9492, 93eqssd 3025 . 2 (𝜑 → dom 𝐵 = 𝐷)
95 df-fn 4955 . 2 ( 𝐵 Fn 𝐷 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝐷))
9616, 94, 95sylanbrc 408 1 (𝜑 𝐵 Fn 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2610  cun 2980  wss 2982  𝒫 cpw 3400  {csn 3416  cop 3419   cuni 3621  Ord word 4145  Oncon0 4146  suc csuc 4148   × cxp 4389  dom cdm 4391  cres 4393  Fun wfun 4946   Fn wfn 4947  wf 4948  cfv 4952  recscrecs 5973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-recs 5974
This theorem is referenced by:  tfr1onlembex  6014  tfr1onlemubacc  6015  tfr1onlemex  6016
  Copyright terms: Public domain W3C validator