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Theorem tfr1onlembfn 6248
Description: Lemma for tfr1on 6254. 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 6246 . . . . 5 (𝜑𝐵𝐴)
1110unissd 3767 . . . 4 (𝜑 𝐵 𝐴)
125, 3tfr1onlemssrecs 6243 . . . 4 (𝜑 𝐴 ⊆ recs(𝐺))
1311, 12sstrd 3111 . . 3 (𝜑 𝐵 ⊆ recs(𝐺))
14 tfrfun 6224 . . 3 Fun recs(𝐺)
15 funss 5149 . . 3 ( 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun 𝐵))
1613, 14, 15mpisyl 1423 . 2 (𝜑 → Fun 𝐵)
17 simpr3 990 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
18 simpl 108 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝜑)
193adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → Ord 𝑋)
20 simpr 109 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝑧𝐷)
218adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝐷𝑋)
2220, 21jca 304 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → (𝑧𝐷𝐷𝑋))
23 ordtr1 4317 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑋 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
2419, 22, 23sylc 62 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝑧𝑋)
2518, 24jca 304 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐷) → (𝜑𝑧𝑋))
262ad2antrr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Fun 𝐺)
273ad2antrr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝑋)
2843adant1r 1210 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑋) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
29283adant1r 1210 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
30 simplr 520 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
31 simpr1 988 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
32 simpr2 989 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
331, 26, 27, 29, 5, 30, 31, 32tfr1onlemsucfn 6244 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
3425, 33sylan 281 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
35 dffn2 5281 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V)
3634, 35sylib 121 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V)
37 fssxp 5297 . . . . . . . . . . . . . . 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 4312 . . . . . . . . . . . . . . . . . . 19 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
403, 8, 39syl2anc 409 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 ∈ On)
41 eloni 4304 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ On → Ord 𝐷)
4240, 41syl 14 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord 𝐷)
4342ad2antrr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝐷)
44 simplr 520 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝐷)
45 ordsucss 4427 . . . . . . . . . . . . . . . 16 (Ord 𝐷 → (𝑧𝐷 → suc 𝑧𝐷))
4643, 44, 45sylc 62 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → suc 𝑧𝐷)
47 xpss1 4656 . . . . . . . . . . . . . . 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 3111 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × V))
50 vex 2692 . . . . . . . . . . . . . . 15 𝑔 ∈ V
51 vex 2692 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
5218adantr 274 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝜑)
5324adantr 274 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
54 simpr1 988 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
55 fneq2 5219 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
5655imbi1d 230 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5756albidv 1797 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5843expia 1184 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
5958alrimiv 1847 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6059ralrimiva 2508 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
61603ad2ant1 1003 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
62 simp2 983 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑧𝑋)
6357, 61, 62rspcdva 2797 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
64 simp3 984 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑔 Fn 𝑧)
65 fneq1 5218 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
66 fveq2 5428 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
6766eleq1d 2209 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
6865, 67imbi12d 233 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
6968spv 1833 . . . . . . . . . . . . . . . . . . 19 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
7063, 64, 69sylc 62 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → (𝐺𝑔) ∈ V)
7152, 53, 54, 70syl3anc 1217 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝐺𝑔) ∈ V)
72 opexg 4157 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
7351, 71, 72sylancr 411 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
74 snexg 4115 . . . . . . . . . . . . . . . 16 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
7573, 74syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
76 unexg 4371 . . . . . . . . . . . . . . 15 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
7750, 75, 76sylancr 411 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
78 elpwg 3522 . . . . . . . . . . . . . 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 166 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × V))
8117, 80eqeltrd 2217 . . . . . . . . . . 11 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ∈ 𝒫 (𝐷 × V))
8281ex 114 . . . . . . . . . 10 ((𝜑𝑧𝐷) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8382exlimdv 1792 . . . . . . . . 9 ((𝜑𝑧𝐷) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8483rexlimdva 2552 . . . . . . . 8 (𝜑 → (∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8584abssdv 3175 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝒫 (𝐷 × V))
866, 85eqsstrid 3147 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝐷 × V))
87 sspwuni 3904 . . . . . 6 (𝐵 ⊆ 𝒫 (𝐷 × V) ↔ 𝐵 ⊆ (𝐷 × V))
8886, 87sylib 121 . . . . 5 (𝜑 𝐵 ⊆ (𝐷 × V))
89 dmss 4745 . . . . 5 ( 𝐵 ⊆ (𝐷 × V) → dom 𝐵 ⊆ dom (𝐷 × V))
9088, 89syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝐷 × V))
91 dmxpss 4976 . . . 4 dom (𝐷 × V) ⊆ 𝐷
9290, 91sstrdi 3113 . . 3 (𝜑 → dom 𝐵𝐷)
931, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembxssdm 6247 . . 3 (𝜑𝐷 ⊆ dom 𝐵)
9492, 93eqssd 3118 . 2 (𝜑 → dom 𝐵 = 𝐷)
95 df-fn 5133 . 2 ( 𝐵 Fn 𝐷 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝐷))
9616, 94, 95sylanbrc 414 1 (𝜑 𝐵 Fn 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963  wal 1330   = wceq 1332  wex 1469  wcel 1481  {cab 2126  wral 2417  wrex 2418  Vcvv 2689  cun 3073  wss 3075  𝒫 cpw 3514  {csn 3531  cop 3534   cuni 3743  Ord word 4291  Oncon0 4292  suc csuc 4294   × cxp 4544  dom cdm 4546  cres 4548  Fun wfun 5124   Fn wfn 5125  wf 5126  cfv 5130  recscrecs 6208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-recs 6209
This theorem is referenced by:  tfr1onlembex  6249  tfr1onlemubacc  6250  tfr1onlemex  6251
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