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Theorem tfrcllembfn 6220
Description: Lemma for tfrcl 6227. The union of 𝐵 is a function defined on 𝑥. (Contributed by Jim Kingdon, 25-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f 𝐹 = recs(𝐺)
tfrcl.g (𝜑 → Fun 𝐺)
tfrcl.x (𝜑 → Ord 𝑋)
tfrcl.ex ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
tfrcllemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfrcllembacc.3 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
tfrcllembacc.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfrcllembacc.4 (𝜑𝐷𝑋)
tfrcllembacc.5 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
Assertion
Ref Expression
tfrcllembfn (𝜑 𝐵:𝐷𝑆)
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑦,𝑧   𝐷,𝑓,𝑔,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,,𝑥,𝑦,𝑧   𝐵,𝑔,,𝑧   𝑤,𝐵,𝑔,𝑧   𝐷,,𝑧   ,𝐺,𝑧   𝑤,𝐺,𝑦   𝑆,𝑔,,𝑧   𝑧,𝑋
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑥,𝑦,𝑓)   𝐷(𝑤)   𝑆(𝑤)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑔,)

Proof of Theorem tfrcllembfn
StepHypRef Expression
1 tfrcl.f . . . . . . 7 𝐹 = recs(𝐺)
2 tfrcl.g . . . . . . 7 (𝜑 → Fun 𝐺)
3 tfrcl.x . . . . . . 7 (𝜑 → Ord 𝑋)
4 tfrcl.ex . . . . . . 7 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
5 tfrcllemsucfn.1 . . . . . . 7 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6 tfrcllembacc.3 . . . . . . 7 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7 tfrcllembacc.u . . . . . . 7 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfrcllembacc.4 . . . . . . 7 (𝜑𝐷𝑋)
9 tfrcllembacc.5 . . . . . . 7 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
101, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembacc 6218 . . . . . 6 (𝜑𝐵𝐴)
1110unissd 3728 . . . . 5 (𝜑 𝐵 𝐴)
125, 3tfrcllemssrecs 6215 . . . . 5 (𝜑 𝐴 ⊆ recs(𝐺))
1311, 12sstrd 3075 . . . 4 (𝜑 𝐵 ⊆ recs(𝐺))
14 tfrfun 6183 . . . 4 Fun recs(𝐺)
15 funss 5110 . . . 4 ( 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun 𝐵))
1613, 14, 15mpisyl 1405 . . 3 (𝜑 → Fun 𝐵)
17 simpr3 972 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
18 simpl 108 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝜑)
193adantr 272 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → Ord 𝑋)
20 simpr 109 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝑧𝐷)
218adantr 272 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝐷) → 𝐷𝑋)
2220, 21jca 302 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐷) → (𝑧𝐷𝐷𝑋))
23 ordtr1 4278 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑋 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
2419, 22, 23sylc 62 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐷) → 𝑧𝑋)
2518, 24jca 302 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐷) → (𝜑𝑧𝑋))
262ad2antrr 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Fun 𝐺)
273ad2antrr 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝑋)
2843adant1r 1192 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑋) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
29283adant1r 1192 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
30 simplr 502 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
31 simpr1 970 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔:𝑧𝑆)
32 simpr2 971 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
331, 26, 27, 29, 5, 30, 31, 32tfrcllemsucfn 6216 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑋) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆)
3425, 33sylan 279 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆)
35 fssxp 5258 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (suc 𝑧 × 𝑆))
3634, 35syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (suc 𝑧 × 𝑆))
37 ordelon 4273 . . . . . . . . . . . . . . . . . . . 20 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
383, 8, 37syl2anc 406 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 ∈ On)
39 eloni 4265 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ On → Ord 𝐷)
4038, 39syl 14 . . . . . . . . . . . . . . . . . 18 (𝜑 → Ord 𝐷)
4140ad2antrr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝐷)
42 simplr 502 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝐷)
43 ordsucss 4388 . . . . . . . . . . . . . . . . 17 (Ord 𝐷 → (𝑧𝐷 → suc 𝑧𝐷))
4441, 42, 43sylc 62 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → suc 𝑧𝐷)
45 xpss1 4617 . . . . . . . . . . . . . . . 16 (suc 𝑧𝐷 → (suc 𝑧 × 𝑆) ⊆ (𝐷 × 𝑆))
4644, 45syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (suc 𝑧 × 𝑆) ⊆ (𝐷 × 𝑆))
4736, 46sstrd 3075 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × 𝑆))
48 vex 2661 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
49 vex 2661 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
5018adantr 272 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝜑)
5124adantr 272 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝑋)
52 simpr1 970 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔:𝑧𝑆)
53 feq2 5224 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (𝑓:𝑥𝑆𝑓:𝑧𝑆))
5453imbi1d 230 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
5554albidv 1778 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
5643expia 1166 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
5756alrimiv 1828 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
5857ralrimiva 2480 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∀𝑥𝑋𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
59583ad2ant1 985 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑋𝑔:𝑧𝑆) → ∀𝑥𝑋𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
60 simp2 965 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑋𝑔:𝑧𝑆) → 𝑧𝑋)
6155, 59, 60rspcdva 2766 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔:𝑧𝑆) → ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆))
62 simp3 966 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑋𝑔:𝑧𝑆) → 𝑔:𝑧𝑆)
63 feq1 5223 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → (𝑓:𝑧𝑆𝑔:𝑧𝑆))
64 fveq2 5387 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
6564eleq1d 2184 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺𝑔) ∈ 𝑆))
6663, 65imbi12d 233 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → ((𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆)))
6766spv 1814 . . . . . . . . . . . . . . . . . . . 20 (∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) → (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆))
6861, 62, 67sylc 62 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑋𝑔:𝑧𝑆) → (𝐺𝑔) ∈ 𝑆)
6950, 51, 52, 68syl3anc 1199 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝐺𝑔) ∈ 𝑆)
70 opexg 4118 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
7149, 69, 70sylancr 408 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
72 snexg 4076 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
7371, 72syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
74 unexg 4332 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
7548, 73, 74sylancr 408 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
76 elpwg 3486 . . . . . . . . . . . . . . 15 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × 𝑆) ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × 𝑆)))
7775, 76syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × 𝑆) ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ⊆ (𝐷 × 𝑆)))
7847, 77mpbird 166 . . . . . . . . . . . . 13 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝒫 (𝐷 × 𝑆))
7917, 78eqeltrd 2192 . . . . . . . . . . . 12 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ∈ 𝒫 (𝐷 × 𝑆))
8079ex 114 . . . . . . . . . . 11 ((𝜑𝑧𝐷) → ((𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × 𝑆)))
8180exlimdv 1773 . . . . . . . . . 10 ((𝜑𝑧𝐷) → (∃𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × 𝑆)))
8281rexlimdva 2524 . . . . . . . . 9 (𝜑 → (∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∈ 𝒫 (𝐷 × 𝑆)))
8382abssdv 3139 . . . . . . . 8 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝒫 (𝐷 × 𝑆))
846, 83eqsstrid 3111 . . . . . . 7 (𝜑𝐵 ⊆ 𝒫 (𝐷 × 𝑆))
85 sspwuni 3865 . . . . . . 7 (𝐵 ⊆ 𝒫 (𝐷 × 𝑆) ↔ 𝐵 ⊆ (𝐷 × 𝑆))
8684, 85sylib 121 . . . . . 6 (𝜑 𝐵 ⊆ (𝐷 × 𝑆))
87 dmss 4706 . . . . . 6 ( 𝐵 ⊆ (𝐷 × 𝑆) → dom 𝐵 ⊆ dom (𝐷 × 𝑆))
8886, 87syl 14 . . . . 5 (𝜑 → dom 𝐵 ⊆ dom (𝐷 × 𝑆))
89 dmxpss 4937 . . . . 5 dom (𝐷 × 𝑆) ⊆ 𝐷
9088, 89syl6ss 3077 . . . 4 (𝜑 → dom 𝐵𝐷)
911, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembxssdm 6219 . . . 4 (𝜑𝐷 ⊆ dom 𝐵)
9290, 91eqssd 3082 . . 3 (𝜑 → dom 𝐵 = 𝐷)
93 df-fn 5094 . . 3 ( 𝐵 Fn 𝐷 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝐷))
9416, 92, 93sylanbrc 411 . 2 (𝜑 𝐵 Fn 𝐷)
95 rnss 4737 . . . 4 ( 𝐵 ⊆ (𝐷 × 𝑆) → ran 𝐵 ⊆ ran (𝐷 × 𝑆))
9686, 95syl 14 . . 3 (𝜑 → ran 𝐵 ⊆ ran (𝐷 × 𝑆))
97 rnxpss 4938 . . 3 ran (𝐷 × 𝑆) ⊆ 𝑆
9896, 97syl6ss 3077 . 2 (𝜑 → ran 𝐵𝑆)
99 df-f 5095 . 2 ( 𝐵:𝐷𝑆 ↔ ( 𝐵 Fn 𝐷 ∧ ran 𝐵𝑆))
10094, 98, 99sylanbrc 411 1 (𝜑 𝐵:𝐷𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945  wal 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2391  wrex 2392  Vcvv 2658  cun 3037  wss 3039  𝒫 cpw 3478  {csn 3495  cop 3498   cuni 3704  Ord word 4252  Oncon0 4253  suc csuc 4255   × cxp 4505  dom cdm 4507  ran crn 4508  cres 4509  Fun wfun 5085   Fn wfn 5086  wf 5087  cfv 5091  recscrecs 6167
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfrcllembex  6221  tfrcllemubacc  6222  tfrcllemex  6223
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