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Theorem tfrcllemsucaccv 6023
Description: Lemma for tfrcl 6033. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-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)))}
tfrcllemsucaccv.yx (𝜑𝑌𝑋)
tfrcllemsucaccv.zy (𝜑𝑧𝑌)
tfrcllemsucaccv.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfrcllemsucaccv.gfn (𝜑𝑔:𝑧𝑆)
tfrcllemsucaccv.gacc (𝜑𝑔𝐴)
Assertion
Ref Expression
tfrcllemsucaccv (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
Distinct variable groups:   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥   𝑓,𝑋,𝑥   𝑓,𝑔,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦   𝑧,𝑓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑔)   𝐴(𝑥,𝑦,𝑧,𝑓,𝑔)   𝑆(𝑦,𝑧,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐺(𝑧,𝑔)   𝑋(𝑦,𝑧,𝑔)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑔)

Proof of Theorem tfrcllemsucaccv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 suceq 4185 . . . . 5 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21eleq1d 2151 . . . 4 (𝑥 = 𝑧 → (suc 𝑥𝑋 ↔ suc 𝑧𝑋))
3 tfrcllemsucaccv.u . . . . 5 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
43ralrimiva 2439 . . . 4 (𝜑 → ∀𝑥 𝑋 suc 𝑥𝑋)
5 tfrcllemsucaccv.zy . . . . 5 (𝜑𝑧𝑌)
6 tfrcllemsucaccv.yx . . . . 5 (𝜑𝑌𝑋)
7 elunii 3626 . . . . 5 ((𝑧𝑌𝑌𝑋) → 𝑧 𝑋)
85, 6, 7syl2anc 403 . . . 4 (𝜑𝑧 𝑋)
92, 4, 8rspcdva 2715 . . 3 (𝜑 → suc 𝑧𝑋)
10 tfrcl.f . . . 4 𝐹 = recs(𝐺)
11 tfrcl.g . . . 4 (𝜑 → Fun 𝐺)
12 tfrcl.x . . . 4 (𝜑 → Ord 𝑋)
13 tfrcl.ex . . . 4 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
14 tfrcllemsucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
155, 6jca 300 . . . . 5 (𝜑 → (𝑧𝑌𝑌𝑋))
16 ordtr1 4171 . . . . 5 (Ord 𝑋 → ((𝑧𝑌𝑌𝑋) → 𝑧𝑋))
1712, 15, 16sylc 61 . . . 4 (𝜑𝑧𝑋)
18 tfrcllemsucaccv.gfn . . . 4 (𝜑𝑔:𝑧𝑆)
19 tfrcllemsucaccv.gacc . . . 4 (𝜑𝑔𝐴)
2010, 11, 12, 13, 14, 17, 18, 19tfrcllemsucfn 6022 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆)
21 vex 2613 . . . . . 6 𝑦 ∈ V
2221elsuc 4189 . . . . 5 (𝑦 ∈ suc 𝑧 ↔ (𝑦𝑧𝑦 = 𝑧))
23 vex 2613 . . . . . . . . . . 11 𝑔 ∈ V
24 feq1 5081 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓:𝑥𝑆𝑔:𝑥𝑆))
25 fveq1 5228 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
26 reseq1 4654 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
2726fveq2d 5233 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑔𝑦)))
2825, 27eqeq12d 2097 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
2928ralbidv 2373 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
3024, 29anbi12d 457 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
3130rexbidv 2374 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
3223, 31, 14elab2 2749 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
3319, 32sylib 120 . . . . . . . . 9 (𝜑 → ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
34 simprrr 507 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
35 simprrl 506 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑔:𝑥𝑆)
36 ffn 5097 . . . . . . . . . . . . 13 (𝑔:𝑥𝑆𝑔 Fn 𝑥)
3735, 36syl 14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑔 Fn 𝑥)
3818adantr 270 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑔:𝑧𝑆)
39 ffn 5097 . . . . . . . . . . . . 13 (𝑔:𝑧𝑆𝑔 Fn 𝑧)
4038, 39syl 14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑔 Fn 𝑧)
41 fndmu 5051 . . . . . . . . . . . 12 ((𝑔 Fn 𝑥𝑔 Fn 𝑧) → 𝑥 = 𝑧)
4237, 40, 41syl2anc 403 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑥 = 𝑧)
4342raleqdv 2560 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → (∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
4434, 43mpbid 145 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
4533, 44rexlimddv 2486 . . . . . . . 8 (𝜑 → ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
4645r19.21bi 2454 . . . . . . 7 ((𝜑𝑦𝑧) → (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
47 ordelon 4166 . . . . . . . . . . . . 13 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
4812, 17, 47syl2anc 403 . . . . . . . . . . . 12 (𝜑𝑧 ∈ On)
49 onelon 4167 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
5048, 49sylan 277 . . . . . . . . . . 11 ((𝜑𝑦𝑧) → 𝑦 ∈ On)
51 eloni 4158 . . . . . . . . . . 11 (𝑦 ∈ On → Ord 𝑦)
52 ordirr 4313 . . . . . . . . . . 11 (Ord 𝑦 → ¬ 𝑦𝑦)
5350, 51, 523syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑧) → ¬ 𝑦𝑦)
54 elequ2 1643 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
5554biimpcd 157 . . . . . . . . . . 11 (𝑦𝑧 → (𝑧 = 𝑦𝑦𝑦))
5655adantl 271 . . . . . . . . . 10 ((𝜑𝑦𝑧) → (𝑧 = 𝑦𝑦𝑦))
5753, 56mtod 622 . . . . . . . . 9 ((𝜑𝑦𝑧) → ¬ 𝑧 = 𝑦)
5857neqned 2256 . . . . . . . 8 ((𝜑𝑦𝑧) → 𝑧𝑦)
59 fvunsng 5409 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧𝑦) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝑔𝑦))
6021, 58, 59sylancr 405 . . . . . . 7 ((𝜑𝑦𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝑔𝑦))
61 eloni 4158 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
6248, 61syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
63 ordelss 4162 . . . . . . . . . . 11 ((Ord 𝑧𝑦𝑧) → 𝑦𝑧)
6462, 63sylan 277 . . . . . . . . . 10 ((𝜑𝑦𝑧) → 𝑦𝑧)
65 resabs1 4688 . . . . . . . . . 10 (𝑦𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))
6664, 65syl 14 . . . . . . . . 9 ((𝜑𝑦𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))
6718, 39syl 14 . . . . . . . . . . . 12 (𝜑𝑔 Fn 𝑧)
68 ordirr 4313 . . . . . . . . . . . . 13 (Ord 𝑧 → ¬ 𝑧𝑧)
6962, 68syl 14 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧𝑧)
70 fsnunres 5416 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
7167, 69, 70syl2anc 403 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
7271reseq1d 4659 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔𝑦))
7372adantr 270 . . . . . . . . 9 ((𝜑𝑦𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔𝑦))
7466, 73eqtr3d 2117 . . . . . . . 8 ((𝜑𝑦𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦) = (𝑔𝑦))
7574fveq2d 5233 . . . . . . 7 ((𝜑𝑦𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)) = (𝐺‘(𝑔𝑦)))
7646, 60, 753eqtr4d 2125 . . . . . 6 ((𝜑𝑦𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
77 feq2 5082 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓:𝑥𝑆𝑓:𝑧𝑆))
7877imbi1d 229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
7978albidv 1747 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
80133expia 1141 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
8180alrimiv 1797 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
8281ralrimiva 2439 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
8379, 82, 17rspcdva 2715 . . . . . . . . . 10 (𝜑 → ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆))
84 feq1 5081 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑧𝑆𝑔:𝑧𝑆))
85 fveq2 5229 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
8685eleq1d 2151 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺𝑔) ∈ 𝑆))
8784, 86imbi12d 232 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆)))
8887spv 1783 . . . . . . . . . 10 (∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) → (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆))
8983, 18, 88sylc 61 . . . . . . . . 9 (𝜑 → (𝐺𝑔) ∈ 𝑆)
90 fndm 5049 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
9167, 90syl 14 . . . . . . . . . 10 (𝜑 → dom 𝑔 = 𝑧)
9269, 91neleqtrrd 2181 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
93 fsnunfv 5415 . . . . . . . . 9 ((𝑧𝑌 ∧ (𝐺𝑔) ∈ 𝑆 ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
945, 89, 92, 93syl3anc 1170 . . . . . . . 8 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
9594adantr 270 . . . . . . 7 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
96 simpr 108 . . . . . . . 8 ((𝜑𝑦 = 𝑧) → 𝑦 = 𝑧)
9796fveq2d 5233 . . . . . . 7 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧))
98 reseq2 4655 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧))
9998, 71sylan9eqr 2137 . . . . . . . 8 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦) = 𝑔)
10099fveq2d 5233 . . . . . . 7 ((𝜑𝑦 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)) = (𝐺𝑔))
10195, 97, 1003eqtr4d 2125 . . . . . 6 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
10276, 101jaodan 744 . . . . 5 ((𝜑 ∧ (𝑦𝑧𝑦 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
10322, 102sylan2b 281 . . . 4 ((𝜑𝑦 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
104103ralrimiva 2439 . . 3 (𝜑 → ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
105 feq2 5082 . . . . . 6 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆))
106 raleq 2554 . . . . . 6 (𝑤 = suc 𝑧 → (∀𝑦𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)) ↔ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
107105, 106anbi12d 457 . . . . 5 (𝑤 = suc 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ∧ ∀𝑦𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆 ∧ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))))
108107rspcev 2710 . . . 4 ((suc 𝑧𝑋 ∧ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆 ∧ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))) → ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ∧ ∀𝑦𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
109 feq2 5082 . . . . . 6 (𝑤 = 𝑥 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆))
110 raleq 2554 . . . . . 6 (𝑤 = 𝑥 → (∀𝑦𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)) ↔ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
111109, 110anbi12d 457 . . . . 5 (𝑤 = 𝑥 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ∧ ∀𝑦𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))))
112111cbvrexv 2583 . . . 4 (∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ∧ ∀𝑦𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))) ↔ ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
113108, 112sylib 120 . . 3 ((suc 𝑧𝑋 ∧ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆 ∧ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))) → ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
1149, 20, 104, 113syl12anc 1168 . 2 (𝜑 → ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
115 vex 2613 . . . . . 6 𝑧 ∈ V
116 opexg 4011 . . . . . 6 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
117115, 89, 116sylancr 405 . . . . 5 (𝜑 → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
118 snexg 3976 . . . . 5 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
119117, 118syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
120 unexg 4224 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
12123, 119, 120sylancr 405 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
122 feq1 5081 . . . . . 6 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑓:𝑥𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆))
123 fveq1 5228 . . . . . . . 8 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑓𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦))
124 reseq1 4654 . . . . . . . . 9 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑓𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))
125124fveq2d 5233 . . . . . . . 8 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝐺‘(𝑓𝑦)) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
126123, 125eqeq12d 2097 . . . . . . 7 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
127126ralbidv 2373 . . . . . 6 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
128122, 127anbi12d 457 . . . . 5 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))))
129128rexbidv 2374 . . . 4 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))))
130129, 14elab2g 2748 . . 3 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))))
131121, 130syl 14 . 2 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))))
132114, 131mpbird 165 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 920  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wne 2249  wral 2353  wrex 2354  Vcvv 2610  cun 2980  wss 2982  {csn 3416  cop 3419   cuni 3621  Ord word 4145  Oncon0 4146  suc csuc 4148  dom cdm 4391  cres 4393  Fun wfun 4946   Fn wfn 4947  wf 4948  cfv 4952  recscrecs 5973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960
This theorem is referenced by:  tfrcllembacc  6024  tfrcllemres  6031
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