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Theorem tfr1onlembfn 6370
Description: Lemma for tfr1on 6376. 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 6368 . . . . 5 (𝜑𝐵𝐴)
1110unissd 3848 . . . 4 (𝜑 𝐵 𝐴)
125, 3tfr1onlemssrecs 6365 . . . 4 (𝜑 𝐴 ⊆ recs(𝐺))
1311, 12sstrd 3180 . . 3 (𝜑 𝐵 ⊆ recs(𝐺))
14 tfrfun 6346 . . 3 Fun recs(𝐺)
15 funss 5254 . . 3 ( 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun 𝐵))
1613, 14, 15mpisyl 1457 . 2 (𝜑 → Fun 𝐵)
17 simpr3 1007 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
18 simpl 109 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝜑)
193adantr 276 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → Ord 𝑋)
20 simpr 110 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝑧𝐷)
218adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝐷𝑋)
2220, 21jca 306 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → (𝑧𝐷𝐷𝑋))
23 ordtr1 4406 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑋 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
2419, 22, 23sylc 62 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝑧𝑋)
2518, 24jca 306 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐷) → (𝜑𝑧𝑋))
262ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Fun 𝐺)
273ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝑋)
2843adant1r 1233 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑋) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
29283adant1r 1233 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
30 simplr 528 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
31 simpr1 1005 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
32 simpr2 1006 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
331, 26, 27, 29, 5, 30, 31, 32tfr1onlemsucfn 6366 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑋) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
3425, 33sylan 283 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
35 dffn2 5386 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V)
3634, 35sylib 122 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧⟶V)
37 fssxp 5402 . . . . . . . . . . . . . . 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 4401 . . . . . . . . . . . . . . . . . . 19 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
403, 8, 39syl2anc 411 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 ∈ On)
41 eloni 4393 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ On → Ord 𝐷)
4240, 41syl 14 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord 𝐷)
4342ad2antrr 488 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝐷)
44 simplr 528 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝐷)
45 ordsucss 4521 . . . . . . . . . . . . . . . 16 (Ord 𝐷 → (𝑧𝐷 → suc 𝑧𝐷))
4643, 44, 45sylc 62 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → suc 𝑧𝐷)
47 xpss1 4754 . . . . . . . . . . . . . . 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 3180 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × V))
50 vex 2755 . . . . . . . . . . . . . . 15 𝑔 ∈ V
51 vex 2755 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
5218adantr 276 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝜑)
5324adantr 276 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
54 simpr1 1005 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
55 fneq2 5324 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
5655imbi1d 231 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5756albidv 1835 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
5843expia 1207 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
5958alrimiv 1885 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6059ralrimiva 2563 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
61603ad2ant1 1020 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
62 simp2 1000 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑧𝑋)
6357, 61, 62rspcdva 2861 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
64 simp3 1001 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔 Fn 𝑧) → 𝑔 Fn 𝑧)
65 fneq1 5323 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
66 fveq2 5534 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
6766eleq1d 2258 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
6865, 67imbi12d 234 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
6968spv 1871 . . . . . . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝐺𝑔) ∈ V)
72 opexg 4246 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
7351, 71, 72sylancr 414 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
74 snexg 4202 . . . . . . . . . . . . . . . 16 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
7573, 74syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
76 unexg 4461 . . . . . . . . . . . . . . 15 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
7750, 75, 76sylancr 414 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
78 elpwg 3598 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × V))
8117, 80eqeltrd 2266 . . . . . . . . . . 11 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ∈ 𝒫 (𝐷 × V))
8281ex 115 . . . . . . . . . 10 ((𝜑𝑧𝐷) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8382exlimdv 1830 . . . . . . . . 9 ((𝜑𝑧𝐷) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8483rexlimdva 2607 . . . . . . . 8 (𝜑 → (∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × V)))
8584abssdv 3244 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝒫 (𝐷 × V))
866, 85eqsstrid 3216 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝐷 × V))
87 sspwuni 3986 . . . . . 6 (𝐵 ⊆ 𝒫 (𝐷 × V) ↔ 𝐵 ⊆ (𝐷 × V))
8886, 87sylib 122 . . . . 5 (𝜑 𝐵 ⊆ (𝐷 × V))
89 dmss 4844 . . . . 5 ( 𝐵 ⊆ (𝐷 × V) → dom 𝐵 ⊆ dom (𝐷 × V))
9088, 89syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝐷 × V))
91 dmxpss 5077 . . . 4 dom (𝐷 × V) ⊆ 𝐷
9290, 91sstrdi 3182 . . 3 (𝜑 → dom 𝐵𝐷)
931, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembxssdm 6369 . . 3 (𝜑𝐷 ⊆ dom 𝐵)
9492, 93eqssd 3187 . 2 (𝜑 → dom 𝐵 = 𝐷)
95 df-fn 5238 . 2 ( 𝐵 Fn 𝐷 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝐷))
9616, 94, 95sylanbrc 417 1 (𝜑 𝐵 Fn 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wal 1362   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wral 2468  wrex 2469  Vcvv 2752  cun 3142  wss 3144  𝒫 cpw 3590  {csn 3607  cop 3610   cuni 3824  Ord word 4380  Oncon0 4381  suc csuc 4383   × cxp 4642  dom cdm 4644  cres 4646  Fun wfun 5229   Fn wfn 5230  wf 5231  cfv 5235  recscrecs 6330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-recs 6331
This theorem is referenced by:  tfr1onlembex  6371  tfr1onlemubacc  6372  tfr1onlemex  6373
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