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Theorem tfrcllemres 6353
Description: Lemma for tfr1on 6341. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-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)))}
tfrcllemres.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfrcllemres.yx (𝜑𝑌𝑋)
Assertion
Ref Expression
tfrcllemres (𝜑𝑌 ⊆ dom 𝐹)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑓)   𝐹(𝑥,𝑦,𝑓)

Proof of Theorem tfrcllemres
Dummy variables 𝑔 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.x . . . . . . . . . 10 (𝜑 → Ord 𝑋)
21adantr 276 . . . . . . . . 9 ((𝜑𝑧𝑌) → Ord 𝑋)
3 simpr 110 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑧𝑌)
4 tfrcllemres.yx . . . . . . . . . . 11 (𝜑𝑌𝑋)
54adantr 276 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑌𝑋)
63, 5jca 306 . . . . . . . . 9 ((𝜑𝑧𝑌) → (𝑧𝑌𝑌𝑋))
7 ordtr1 4382 . . . . . . . . 9 (Ord 𝑋 → ((𝑧𝑌𝑌𝑋) → 𝑧𝑋))
82, 6, 7sylc 62 . . . . . . . 8 ((𝜑𝑧𝑌) → 𝑧𝑋)
9 tfrcl.f . . . . . . . . 9 𝐹 = recs(𝐺)
10 tfrcl.g . . . . . . . . 9 (𝜑 → Fun 𝐺)
11 tfrcl.ex . . . . . . . . 9 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
12 tfrcllemsucfn.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
13 tfrcllemres.u . . . . . . . . 9 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
149, 10, 1, 11, 12, 13tfrcllemaccex 6352 . . . . . . . 8 ((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
158, 14syldan 282 . . . . . . 7 ((𝜑𝑧𝑌) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1610ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Fun 𝐺)
171ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Ord 𝑋)
18113adant1r 1231 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
19183adant1r 1231 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
204ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑌𝑋)
213adantr 276 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑧𝑌)
2213adantlr 477 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
2322adantlr 477 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
24 simprl 529 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔:𝑧𝑆)
25 feq2 5341 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑔:𝑥𝑆𝑔:𝑧𝑆))
26 raleq 2670 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
2725, 26anbi12d 473 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) ↔ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
28 fveq2 5507 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (𝑔𝑦) = (𝑔𝑢))
29 reseq2 4895 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → (𝑔𝑦) = (𝑔𝑢))
3029fveq2d 5511 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (𝐺‘(𝑔𝑦)) = (𝐺‘(𝑔𝑢)))
3128, 30eqeq12d 2190 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → ((𝑔𝑦) = (𝐺‘(𝑔𝑦)) ↔ (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
3231cbvralv 2701 . . . . . . . . . . . . . 14 (∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
3332anbi2i 457 . . . . . . . . . . . . 13 ((𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) ↔ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
3427, 33bitrdi 196 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) ↔ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
3534rspcev 2839 . . . . . . . . . . 11 ((𝑧𝑋 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
368, 35sylan 283 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
37 vex 2738 . . . . . . . . . . 11 𝑔 ∈ V
38 feq1 5340 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓:𝑥𝑆𝑔:𝑥𝑆))
39 fveq1 5506 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
40 reseq1 4894 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
4140fveq2d 5511 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑔𝑦)))
4239, 41eqeq12d 2190 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
4342ralbidv 2475 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
4438, 43anbi12d 473 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
4544rexbidv 2476 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
4637, 45, 12elab2 2883 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
4736, 46sylibr 134 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔𝐴)
489, 16, 17, 19, 12, 20, 21, 23, 24, 47tfrcllemsucaccv 6345 . . . . . . . 8 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
49 vex 2738 . . . . . . . . . . 11 𝑧 ∈ V
5025imbi1d 231 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑔:𝑥𝑆 → (𝐺𝑔) ∈ 𝑆) ↔ (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆)))
51113expia 1205 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
5251alrimiv 1872 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
53 fveq2 5507 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
5453eleq1d 2244 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺𝑔) ∈ 𝑆))
5538, 54imbi12d 234 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔 → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑔:𝑥𝑆 → (𝐺𝑔) ∈ 𝑆)))
5655spv 1858 . . . . . . . . . . . . . . . . 17 (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) → (𝑔:𝑥𝑆 → (𝐺𝑔) ∈ 𝑆))
5752, 56syl 14 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑔:𝑥𝑆 → (𝐺𝑔) ∈ 𝑆))
5857ralrimiva 2548 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑋 (𝑔:𝑥𝑆 → (𝐺𝑔) ∈ 𝑆))
5958adantr 276 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑌) → ∀𝑥𝑋 (𝑔:𝑥𝑆 → (𝐺𝑔) ∈ 𝑆))
6050, 59, 8rspcdva 2844 . . . . . . . . . . . . 13 ((𝜑𝑧𝑌) → (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆))
6160imp 124 . . . . . . . . . . . 12 (((𝜑𝑧𝑌) ∧ 𝑔:𝑧𝑆) → (𝐺𝑔) ∈ 𝑆)
6224, 61syldan 282 . . . . . . . . . . 11 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝐺𝑔) ∈ 𝑆)
63 opexg 4222 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
6449, 62, 63sylancr 414 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
65 snidg 3618 . . . . . . . . . 10 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → ⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩})
66 elun2 3301 . . . . . . . . . 10 (⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩} → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
6764, 65, 663syl 17 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
68 opeldmg 4825 . . . . . . . . . 10 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ 𝑆) → (⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
6949, 62, 68sylancr 414 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
7067, 69mpd 13 . . . . . . . 8 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
71 dmeq 4820 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
7271eleq2d 2245 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
7372rspcev 2839 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
7448, 70, 73syl2anc 411 . . . . . . 7 (((𝜑𝑧𝑌) ∧ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
7515, 74exlimddv 1896 . . . . . 6 ((𝜑𝑧𝑌) → ∃𝐴 𝑧 ∈ dom )
76 eliun 3886 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
7775, 76sylibr 134 . . . . 5 ((𝜑𝑧𝑌) → 𝑧 𝐴 dom )
7877ex 115 . . . 4 (𝜑 → (𝑧𝑌𝑧 𝐴 dom ))
7978ssrdv 3159 . . 3 (𝜑𝑌 𝐴 dom )
80 dmuni 4830 . . . 4 dom 𝐴 = 𝐴 dom
8112, 1tfrcllemssrecs 6343 . . . . 5 (𝜑 𝐴 ⊆ recs(𝐺))
82 dmss 4819 . . . . 5 ( 𝐴 ⊆ recs(𝐺) → dom 𝐴 ⊆ dom recs(𝐺))
8381, 82syl 14 . . . 4 (𝜑 → dom 𝐴 ⊆ dom recs(𝐺))
8480, 83eqsstrrid 3200 . . 3 (𝜑 𝐴 dom ⊆ dom recs(𝐺))
8579, 84sstrd 3163 . 2 (𝜑𝑌 ⊆ dom recs(𝐺))
869dmeqi 4821 . 2 dom 𝐹 = dom recs(𝐺)
8785, 86sseqtrrdi 3202 1 (𝜑𝑌 ⊆ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wal 1351   = wceq 1353  wex 1490  wcel 2146  {cab 2161  wral 2453  wrex 2454  Vcvv 2735  cun 3125  wss 3127  {csn 3589  cop 3592   cuni 3805   ciun 3882  Ord word 4356  suc csuc 4359  dom cdm 4620  cres 4622  Fun wfun 5202  wf 5204  cfv 5208  recscrecs 6295
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-recs 6296
This theorem is referenced by:  tfrcldm  6354
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