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Theorem tfrcllembxssdm 6351
Description: Lemma for tfrcl 6359. The union of 𝐵 is defined on all elements of 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
tfrcllembxssdm (𝜑𝐷 ⊆ dom 𝐵)
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,,𝑧   𝑧,𝑋
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑥,𝑦,𝑓)   𝐷(𝑤)   𝑆(𝑤,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑔,)

Proof of Theorem tfrcllembxssdm
StepHypRef Expression
1 tfrcllembacc.5 . . . 4 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
2 fveq2 5511 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑔𝑤) = (𝑔𝑦))
3 reseq2 4898 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑔𝑤) = (𝑔𝑦))
43fveq2d 5515 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐺‘(𝑔𝑤)) = (𝐺‘(𝑔𝑦)))
52, 4eqeq12d 2192 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑔𝑤) = (𝐺‘(𝑔𝑤)) ↔ (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
65cbvralv 2703 . . . . . . 7 (∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
76anbi2i 457 . . . . . 6 ((𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))) ↔ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
87exbii 1605 . . . . 5 (∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))) ↔ ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
98ralbii 2483 . . . 4 (∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))) ↔ ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
101, 9sylib 122 . . 3 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
11 simp1 997 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝜑)
12 simp2 998 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝑧𝐷)
13 tfrcllembacc.4 . . . . . . . . . 10 (𝜑𝐷𝑋)
1411, 13syl 14 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝐷𝑋)
15 tfrcl.x . . . . . . . . . . 11 (𝜑 → Ord 𝑋)
16 ordtr1 4385 . . . . . . . . . . 11 (Ord 𝑋 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
1715, 16syl 14 . . . . . . . . . 10 (𝜑 → ((𝑧𝐷𝐷𝑋) → 𝑧𝑋))
1817imp 124 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐷𝐷𝑋)) → 𝑧𝑋)
1911, 12, 14, 18syl12anc 1236 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝑧𝑋)
20 simp3l 1025 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝑔:𝑧𝑆)
21 feq2 5345 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓:𝑥𝑆𝑓:𝑧𝑆))
2221imbi1d 231 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
2322albidv 1824 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
24 tfrcl.ex . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
25243expia 1205 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
2625alrimiv 1874 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
2726ralrimiva 2550 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑋𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
2827adantr 276 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → ∀𝑥𝑋𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
29 simpr 110 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 𝑧𝑋)
3023, 28, 29rspcdva 2846 . . . . . . . . . 10 ((𝜑𝑧𝑋) → ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆))
31 feq1 5344 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑧𝑆𝑔:𝑧𝑆))
32 fveq2 5511 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
3332eleq1d 2246 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺𝑔) ∈ 𝑆))
3431, 33imbi12d 234 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆)))
3534spv 1860 . . . . . . . . . 10 (∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) → (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆))
3630, 35syl 14 . . . . . . . . 9 ((𝜑𝑧𝑋) → (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆))
3736imp 124 . . . . . . . 8 (((𝜑𝑧𝑋) ∧ 𝑔:𝑧𝑆) → (𝐺𝑔) ∈ 𝑆)
3811, 19, 20, 37syl21anc 1237 . . . . . . 7 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → (𝐺𝑔) ∈ 𝑆)
39 vex 2740 . . . . . . . . . 10 𝑧 ∈ V
40 opexg 4225 . . . . . . . . . 10 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
4139, 38, 40sylancr 414 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
42 snidg 3620 . . . . . . . . 9 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → ⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩})
43 elun2 3303 . . . . . . . . 9 (⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩} → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
4441, 42, 433syl 17 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
45 simp3r 1026 . . . . . . . . . . . . 13 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
46 rspe 2526 . . . . . . . . . . . . 13 ((𝑧𝑋 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ∃𝑧𝑋 (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
4719, 20, 45, 46syl12anc 1236 . . . . . . . . . . . 12 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ∃𝑧𝑋 (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
48 feq2 5345 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑔:𝑧𝑆𝑔:𝑥𝑆))
49 raleq 2672 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
5048, 49anbi12d 473 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) ↔ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
5150cbvrexv 2704 . . . . . . . . . . . 12 (∃𝑧𝑋 (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
5247, 51sylib 122 . . . . . . . . . . 11 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
53 vex 2740 . . . . . . . . . . . 12 𝑔 ∈ V
54 feq1 5344 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑓:𝑥𝑆𝑔:𝑥𝑆))
55 fveq1 5510 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
56 reseq1 4897 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
5756fveq2d 5515 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔 → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑔𝑦)))
5855, 57eqeq12d 2192 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
5958ralbidv 2477 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
6054, 59anbi12d 473 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
6160rexbidv 2478 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
62 tfrcllemsucfn.1 . . . . . . . . . . . 12 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6353, 61, 62elab2 2885 . . . . . . . . . . 11 (𝑔𝐴 ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
6452, 63sylibr 134 . . . . . . . . . 10 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝑔𝐴)
6512, 20, 643jca 1177 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → (𝑧𝐷𝑔:𝑧𝑆𝑔𝐴))
66 snexg 4181 . . . . . . . . . . 11 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
67 unexg 4440 . . . . . . . . . . . 12 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
6853, 67mpan 424 . . . . . . . . . . 11 ({⟨𝑧, (𝐺𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
6941, 66, 683syl 17 . . . . . . . . . 10 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
70 isset 2743 . . . . . . . . . 10 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V ↔ ∃ = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
7169, 70sylib 122 . . . . . . . . 9 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ∃ = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
72 simpr3 1005 . . . . . . . . . . . . 13 ((𝑧𝐷 ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
73 19.8a 1590 . . . . . . . . . . . . . 14 ((𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∃𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
74 rspe 2526 . . . . . . . . . . . . . . 15 ((𝑧𝐷 ∧ ∃𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
75 tfrcllembacc.3 . . . . . . . . . . . . . . . 16 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7675abeq2i 2288 . . . . . . . . . . . . . . 15 (𝐵 ↔ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
7774, 76sylibr 134 . . . . . . . . . . . . . 14 ((𝑧𝐷 ∧ ∃𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐵)
7873, 77sylan2 286 . . . . . . . . . . . . 13 ((𝑧𝐷 ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐵)
7972, 78eqeltrrd 2255 . . . . . . . . . . . 12 ((𝑧𝐷 ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵)
80793exp2 1225 . . . . . . . . . . 11 (𝑧𝐷 → (𝑔:𝑧𝑆 → (𝑔𝐴 → ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵))))
81803imp 1193 . . . . . . . . . 10 ((𝑧𝐷𝑔:𝑧𝑆𝑔𝐴) → ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵))
8281exlimdv 1819 . . . . . . . . 9 ((𝑧𝐷𝑔:𝑧𝑆𝑔𝐴) → (∃ = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵))
8365, 71, 82sylc 62 . . . . . . . 8 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵)
84 elunii 3812 . . . . . . . 8 ((⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵)
8544, 83, 84syl2anc 411 . . . . . . 7 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵)
86 opeq2 3777 . . . . . . . . . 10 (𝑤 = (𝐺𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐺𝑔)⟩)
8786eleq1d 2246 . . . . . . . . 9 (𝑤 = (𝐺𝑔) → (⟨𝑧, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵))
8887spcegv 2825 . . . . . . . 8 ((𝐺𝑔) ∈ 𝑆 → (⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵 → ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵))
8939eldm2 4821 . . . . . . . 8 (𝑧 ∈ dom 𝐵 ↔ ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵)
9088, 89syl6ibr 162 . . . . . . 7 ((𝐺𝑔) ∈ 𝑆 → (⟨𝑧, (𝐺𝑔)⟩ ∈ 𝐵𝑧 ∈ dom 𝐵))
9138, 85, 90sylc 62 . . . . . 6 ((𝜑𝑧𝐷 ∧ (𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))) → 𝑧 ∈ dom 𝐵)
92913expia 1205 . . . . 5 ((𝜑𝑧𝐷) → ((𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) → 𝑧 ∈ dom 𝐵))
9392exlimdv 1819 . . . 4 ((𝜑𝑧𝐷) → (∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) → 𝑧 ∈ dom 𝐵))
9493ralimdva 2544 . . 3 (𝜑 → (∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))) → ∀𝑧𝐷 𝑧 ∈ dom 𝐵))
9510, 94mpd 13 . 2 (𝜑 → ∀𝑧𝐷 𝑧 ∈ dom 𝐵)
96 dfss3 3145 . 2 (𝐷 ⊆ dom 𝐵 ↔ ∀𝑧𝐷 𝑧 ∈ dom 𝐵)
9795, 96sylibr 134 1 (𝜑𝐷 ⊆ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wal 1351   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2737  cun 3127  wss 3129  {csn 3591  cop 3594   cuni 3807  Ord word 4359  suc csuc 4362  dom cdm 4623  cres 4625  Fun wfun 5206  wf 5208  cfv 5212  recscrecs 6299
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-tr 4099  df-iord 4363  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220
This theorem is referenced by:  tfrcllembfn  6352
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