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Theorem tfr1onlembfn 6063
Description: Lemma for tfr1on 6069. 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 6061 . . . . 5 (𝜑𝐵𝐴)
1110unissd 3660 . . . 4 (𝜑 𝐵 𝐴)
125, 3tfr1onlemssrecs 6058 . . . 4 (𝜑 𝐴 ⊆ recs(𝐺))
1311, 12sstrd 3024 . . 3 (𝜑 𝐵 ⊆ recs(𝐺))
14 tfrfun 6039 . . 3 Fun recs(𝐺)
15 funss 4999 . . 3 ( 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun 𝐵))
1613, 14, 15mpisyl 1378 . 2 (𝜑 → Fun 𝐵)
17 simpr3 949 . . . . . . . . . . . 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 4189 . . . . . . . . . . . . . . . . . . 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 1165 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑋) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
29283adant1r 1165 . . . . . . . . . . . . . . . . . 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 947 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
32 simpr2 948 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
331, 26, 27, 29, 5, 30, 31, 32tfr1onlemsucfn 6059 . . . . . . . . . . . . . . . . 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 5128 . . . . . . . . . . . . . . . 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 5142 . . . . . . . . . . . . . . 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 4184 . . . . . . . . . . . . . . . . . . 19 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
403, 8, 39syl2anc 403 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 ∈ On)
41 eloni 4176 . . . . . . . . . . . . . . . . . 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 4294 . . . . . . . . . . . . . . . 16 (Ord 𝐷 → (𝑧𝐷 → suc 𝑧𝐷))
4643, 44, 45sylc 61 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → suc 𝑧𝐷)
47 xpss1 4516 . . . . . . . . . . . . . . 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 3024 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × V))
50 vex 2618 . . . . . . . . . . . . . . 15 𝑔 ∈ V
51 vex 2618 . . . . . . . . . . . . . . . . 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 947 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
55 fneq2 5068 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
5655imbi1d 229 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5756albidv 1749 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5843expia 1143 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
5958alrimiv 1799 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6059ralrimiva 2442 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
61603ad2ant1 962 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
62 simp2 942 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑧𝑋)
6357, 61, 62rspcdva 2720 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
64 simp3 943 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑔 Fn 𝑧)
65 fneq1 5067 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
66 fveq2 5268 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
6766eleq1d 2153 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
6865, 67imbi12d 232 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
6968spv 1785 . . . . . . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝐺𝑔) ∈ V)
72 opexg 4029 . . . . . . . . . . . . . . . . 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 3993 . . . . . . . . . . . . . . . 16 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
7573, 74syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
76 unexg 4242 . . . . . . . . . . . . . . 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 3423 . . . . . . . . . . . . . 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 2161 . . . . . . . . . . 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 1744 . . . . . . . . 9 ((𝜑𝑧𝐷) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8483rexlimdva 2485 . . . . . . . 8 (𝜑 → (∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8584abssdv 3084 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝒫 (𝐷 × V))
866, 85syl5eqss 3059 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝐷 × V))
87 sspwuni 3795 . . . . . 6 (𝐵 ⊆ 𝒫 (𝐷 × V) ↔ 𝐵 ⊆ (𝐷 × V))
8886, 87sylib 120 . . . . 5 (𝜑 𝐵 ⊆ (𝐷 × V))
89 dmss 4603 . . . . 5 ( 𝐵 ⊆ (𝐷 × V) → dom 𝐵 ⊆ dom (𝐷 × V))
9088, 89syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝐷 × V))
91 dmxpss 4827 . . . 4 dom (𝐷 × V) ⊆ 𝐷
9290, 91syl6ss 3026 . . 3 (𝜑 → dom 𝐵𝐷)
931, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembxssdm 6062 . . 3 (𝜑𝐷 ⊆ dom 𝐵)
9492, 93eqssd 3031 . 2 (𝜑 → dom 𝐵 = 𝐷)
95 df-fn 4984 . 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 922  wal 1285   = wceq 1287  wex 1424  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2615  cun 2986  wss 2988  𝒫 cpw 3415  {csn 3431  cop 3434   cuni 3636  Ord word 4163  Oncon0 4164  suc csuc 4166   × cxp 4409  dom cdm 4411  cres 4413  Fun wfun 4975   Fn wfn 4976  wf 4977  cfv 4981  recscrecs 6023
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-recs 6024
This theorem is referenced by:  tfr1onlembex  6064  tfr1onlemubacc  6065  tfr1onlemex  6066
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