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Theorem tfr1onlembxssdm 6292
Description: Lemma for tfr1on 6299. The union of 𝐵 is defined on all elements of 𝑋. (Contributed by Jim Kingdon, 14-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
tfr1onlembxssdm (𝜑𝐷 ⊆ dom 𝐵)
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,𝐵,𝑔,𝑧   𝐷,,𝑧   ,𝐺,𝑧   𝑤,𝐺,𝑓,𝑥,𝑦   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 tfr1onlembxssdm
StepHypRef Expression
1 tfr1onlembacc.5 . . 3 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
2 simp1 982 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝜑)
3 simp2 983 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝑧𝐷)
4 tfr1onlembacc.4 . . . . . . . . . 10 (𝜑𝐷𝑋)
52, 4syl 14 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝐷𝑋)
6 tfr1on.x . . . . . . . . . . 11 (𝜑 → Ord 𝑋)
7 ordtr1 4350 . . . . . . . . . . 11 (Ord 𝑋 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
86, 7syl 14 . . . . . . . . . 10 (𝜑 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
98imp 123 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐷𝐷𝑋)) → 𝑧𝑋)
102, 3, 5, 9syl12anc 1218 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝑧𝑋)
11 simp3l 1010 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝑔 Fn 𝑧)
12 fneq2 5261 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
1312imbi1d 230 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
1413albidv 1804 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
15 tfr1on.ex . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
16153expia 1187 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
1716alrimiv 1854 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
1817ralrimiva 2530 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
1918adantr 274 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
20 simpr 109 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 𝑧𝑋)
2114, 19, 20rspcdva 2821 . . . . . . . . . 10 ((𝜑𝑧𝑋) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
22 fneq1 5260 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
23 fveq2 5470 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
2423eleq1d 2226 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
2522, 24imbi12d 233 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
2625spv 1840 . . . . . . . . . 10 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
2721, 26syl 14 . . . . . . . . 9 ((𝜑𝑧𝑋) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
2827imp 123 . . . . . . . 8 (((𝜑𝑧𝑋) ∧ 𝑔 Fn 𝑧) → (𝐺𝑔) ∈ V)
292, 10, 11, 28syl21anc 1219 . . . . . . 7 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → (𝐺𝑔) ∈ V)
30 vex 2715 . . . . . . . . . 10 𝑧 ∈ V
31 opexg 4190 . . . . . . . . . 10 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
3230, 29, 31sylancr 411 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
33 snidg 3590 . . . . . . . . 9 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → ⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩})
34 elun2 3276 . . . . . . . . 9 (⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩} → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
3532, 33, 343syl 17 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
36 simp3r 1011 . . . . . . . . . . . 12 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))
37 rspe 2506 . . . . . . . . . . . 12 ((𝑧𝑋 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ∃𝑧𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
3810, 11, 36, 37syl12anc 1218 . . . . . . . . . . 11 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ∃𝑧𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
39 vex 2715 . . . . . . . . . . . 12 𝑔 ∈ V
40 tfr1onlemsucfn.1 . . . . . . . . . . . . 13 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
4140tfr1onlem3ag 6286 . . . . . . . . . . . 12 (𝑔 ∈ V → (𝑔𝐴 ↔ ∃𝑧𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))))
4239, 41ax-mp 5 . . . . . . . . . . 11 (𝑔𝐴 ↔ ∃𝑧𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
4338, 42sylibr 133 . . . . . . . . . 10 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝑔𝐴)
443, 11, 433jca 1162 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → (𝑧𝐷𝑔 Fn 𝑧𝑔𝐴))
45 snexg 4147 . . . . . . . . . . 11 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
46 unexg 4405 . . . . . . . . . . . 12 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
4739, 46mpan 421 . . . . . . . . . . 11 ({⟨𝑧, (𝐺𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
4832, 45, 473syl 17 . . . . . . . . . 10 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
49 isset 2718 . . . . . . . . . 10 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V ↔ ∃ = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
5048, 49sylib 121 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ∃ = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
51 simpr3 990 . . . . . . . . . . . . 13 ((𝑧𝐷 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
52 19.8a 1570 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
53 rspe 2506 . . . . . . . . . . . . . . 15 ((𝑧𝐷 ∧ ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
54 tfr1onlembacc.3 . . . . . . . . . . . . . . . 16 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
5554abeq2i 2268 . . . . . . . . . . . . . . 15 (𝐵 ↔ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
5653, 55sylibr 133 . . . . . . . . . . . . . 14 ((𝑧𝐷 ∧ ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐵)
5752, 56sylan2 284 . . . . . . . . . . . . 13 ((𝑧𝐷 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐵)
5851, 57eqeltrrd 2235 . . . . . . . . . . . 12 ((𝑧𝐷 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵)
59583exp2 1207 . . . . . . . . . . 11 (𝑧𝐷 → (𝑔 Fn 𝑧 → (𝑔𝐴 → ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵))))
60593imp 1176 . . . . . . . . . 10 ((𝑧𝐷𝑔 Fn 𝑧𝑔𝐴) → ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵))
6160exlimdv 1799 . . . . . . . . 9 ((𝑧𝐷𝑔 Fn 𝑧𝑔𝐴) → (∃ = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵))
6244, 50, 61sylc 62 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵)
63 elunii 3779 . . . . . . . 8 ((⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵)
6435, 62, 63syl2anc 409 . . . . . . 7 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵)
65 opeq2 3744 . . . . . . . . . 10 (𝑤 = (𝐺𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐺𝑔)⟩)
6665eleq1d 2226 . . . . . . . . 9 (𝑤 = (𝐺𝑔) → (⟨𝑧, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵))
6766spcegv 2800 . . . . . . . 8 ((𝐺𝑔) ∈ V → (⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵 → ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵))
6830eldm2 4786 . . . . . . . 8 (𝑧 ∈ dom 𝐵 ↔ ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵)
6967, 68syl6ibr 161 . . . . . . 7 ((𝐺𝑔) ∈ V → (⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵𝑧 ∈ dom 𝐵))
7029, 64, 69sylc 62 . . . . . 6 ((𝜑𝑧𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))) → 𝑧 ∈ dom 𝐵)
71703expia 1187 . . . . 5 ((𝜑𝑧𝐷) → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
7271exlimdv 1799 . . . 4 ((𝜑𝑧𝐷) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
7372ralimdva 2524 . . 3 (𝜑 → (∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))) → ∀𝑧𝐷 𝑧 ∈ dom 𝐵))
741, 73mpd 13 . 2 (𝜑 → ∀𝑧𝐷 𝑧 ∈ dom 𝐵)
75 dfss3 3118 . 2 (𝐷 ⊆ dom 𝐵 ↔ ∀𝑧𝐷 𝑧 ∈ dom 𝐵)
7674, 75sylibr 133 1 (𝜑𝐷 ⊆ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963  wal 1333   = wceq 1335  wex 1472  wcel 2128  {cab 2143  wral 2435  wrex 2436  Vcvv 2712  cun 3100  wss 3102  {csn 3561  cop 3564   cuni 3774  Ord word 4324  suc csuc 4327  dom cdm 4588  cres 4590  Fun wfun 5166   Fn wfn 5167  cfv 5172  recscrecs 6253
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-tr 4065  df-iord 4328  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-res 4600  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180
This theorem is referenced by:  tfr1onlembfn  6293
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