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Theorem tfrcllemsucaccv 6101
Description: Lemma for tfrcl 6111. 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 4220 . . . . 5 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21eleq1d 2156 . . . 4 (𝑥 = 𝑧 → (suc 𝑥𝑋 ↔ suc 𝑧𝑋))
3 tfrcllemsucaccv.u . . . . 5 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
43ralrimiva 2446 . . . 4 (𝜑 → ∀𝑥 𝑋 suc 𝑥𝑋)
5 tfrcllemsucaccv.zy . . . . 5 (𝜑𝑧𝑌)
6 tfrcllemsucaccv.yx . . . . 5 (𝜑𝑌𝑋)
7 elunii 3653 . . . . 5 ((𝑧𝑌𝑌𝑋) → 𝑧 𝑋)
85, 6, 7syl2anc 403 . . . 4 (𝜑𝑧 𝑋)
92, 4, 8rspcdva 2727 . . 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 4206 . . . . 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 6100 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆)
21 vex 2622 . . . . . 6 𝑦 ∈ V
2221elsuc 4224 . . . . 5 (𝑦 ∈ suc 𝑧 ↔ (𝑦𝑧𝑦 = 𝑧))
23 vex 2622 . . . . . . . . . . 11 𝑔 ∈ V
24 feq1 5131 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓:𝑥𝑆𝑔:𝑥𝑆))
25 fveq1 5288 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
26 reseq1 4695 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
2726fveq2d 5293 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑔𝑦)))
2825, 27eqeq12d 2102 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑔𝑦) = (𝐺‘(𝑔𝑦))))
2928ralbidv 2380 . . . . . . . . . . . . 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 2381 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))))
3223, 31, 14elab2 2761 . . . . . . . . . 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 5147 . . . . . . . . . . . . 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 5147 . . . . . . . . . . . . 13 (𝑔:𝑧𝑆𝑔 Fn 𝑧)
4038, 39syl 14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑔 Fn 𝑧)
41 fndmu 5101 . . . . . . . . . . . 12 ((𝑔 Fn 𝑥𝑔 Fn 𝑧) → 𝑥 = 𝑧)
4237, 40, 41syl2anc 403 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑔:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐺‘(𝑔𝑦))))) → 𝑥 = 𝑧)
4342raleqdv 2568 . . . . . . . . . 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 2493 . . . . . . . 8 (𝜑 → ∀𝑦𝑧 (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
4645r19.21bi 2461 . . . . . . 7 ((𝜑𝑦𝑧) → (𝑔𝑦) = (𝐺‘(𝑔𝑦)))
47 ordelon 4201 . . . . . . . . . . . . 13 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
4812, 17, 47syl2anc 403 . . . . . . . . . . . 12 (𝜑𝑧 ∈ On)
49 onelon 4202 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
5048, 49sylan 277 . . . . . . . . . . 11 ((𝜑𝑦𝑧) → 𝑦 ∈ On)
51 eloni 4193 . . . . . . . . . . 11 (𝑦 ∈ On → Ord 𝑦)
52 ordirr 4348 . . . . . . . . . . 11 (Ord 𝑦 → ¬ 𝑦𝑦)
5350, 51, 523syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑧) → ¬ 𝑦𝑦)
54 elequ2 1648 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
5554biimpcd 157 . . . . . . . . . . 11 (𝑦𝑧 → (𝑧 = 𝑦𝑦𝑦))
5655adantl 271 . . . . . . . . . 10 ((𝜑𝑦𝑧) → (𝑧 = 𝑦𝑦𝑦))
5753, 56mtod 624 . . . . . . . . 9 ((𝜑𝑦𝑧) → ¬ 𝑧 = 𝑦)
5857neqned 2262 . . . . . . . 8 ((𝜑𝑦𝑧) → 𝑧𝑦)
59 fvunsng 5475 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧𝑦) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝑔𝑦))
6021, 58, 59sylancr 405 . . . . . . 7 ((𝜑𝑦𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝑔𝑦))
61 eloni 4193 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
6248, 61syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
63 ordelss 4197 . . . . . . . . . . 11 ((Ord 𝑧𝑦𝑧) → 𝑦𝑧)
6462, 63sylan 277 . . . . . . . . . 10 ((𝜑𝑦𝑧) → 𝑦𝑧)
65 resabs1 4729 . . . . . . . . . 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 4348 . . . . . . . . . . . . 13 (Ord 𝑧 → ¬ 𝑧𝑧)
6962, 68syl 14 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧𝑧)
70 fsnunres 5482 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
7167, 69, 70syl2anc 403 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
7271reseq1d 4700 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔𝑦))
7372adantr 270 . . . . . . . . 9 ((𝜑𝑦𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔𝑦))
7466, 73eqtr3d 2122 . . . . . . . 8 ((𝜑𝑦𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦) = (𝑔𝑦))
7574fveq2d 5293 . . . . . . 7 ((𝜑𝑦𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)) = (𝐺‘(𝑔𝑦)))
7646, 60, 753eqtr4d 2130 . . . . . 6 ((𝜑𝑦𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
77 feq2 5132 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓:𝑥𝑆𝑓:𝑧𝑆))
7877imbi1d 229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
7978albidv 1752 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆)))
80133expia 1145 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
8180alrimiv 1802 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
8281ralrimiva 2446 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
8379, 82, 17rspcdva 2727 . . . . . . . . . 10 (𝜑 → ∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆))
84 feq1 5131 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑧𝑆𝑔:𝑧𝑆))
85 fveq2 5289 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
8685eleq1d 2156 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺𝑔) ∈ 𝑆))
8784, 86imbi12d 232 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆)))
8887spv 1788 . . . . . . . . . 10 (∀𝑓(𝑓:𝑧𝑆 → (𝐺𝑓) ∈ 𝑆) → (𝑔:𝑧𝑆 → (𝐺𝑔) ∈ 𝑆))
8983, 18, 88sylc 61 . . . . . . . . 9 (𝜑 → (𝐺𝑔) ∈ 𝑆)
90 fndm 5099 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
9167, 90syl 14 . . . . . . . . . 10 (𝜑 → dom 𝑔 = 𝑧)
9269, 91neleqtrrd 2186 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
93 fsnunfv 5481 . . . . . . . . 9 ((𝑧𝑌 ∧ (𝐺𝑔) ∈ 𝑆 ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
945, 89, 92, 93syl3anc 1174 . . . . . . . 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 5293 . . . . . . 7 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧))
98 reseq2 4696 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧))
9998, 71sylan9eqr 2142 . . . . . . . 8 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦) = 𝑔)
10099fveq2d 5293 . . . . . . 7 ((𝜑𝑦 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)) = (𝐺𝑔))
10195, 97, 1003eqtr4d 2130 . . . . . 6 ((𝜑𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
10276, 101jaodan 746 . . . . 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 2446 . . 3 (𝜑 → ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
105 feq2 5132 . . . . . 6 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆))
106 raleq 2562 . . . . . 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 2722 . . . 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 5132 . . . . . 6 (𝑤 = 𝑥 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑤𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆))
110 raleq 2562 . . . . . 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 2591 . . . 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 1172 . 2 (𝜑 → ∃𝑥𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆 ∧ ∀𝑦𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
115 vex 2622 . . . . . 6 𝑧 ∈ V
116 opexg 4046 . . . . . 6 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
117115, 89, 116sylancr 405 . . . . 5 (𝜑 → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
118 snexg 4010 . . . . 5 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
119117, 118syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
120 unexg 4259 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
12123, 119, 120sylancr 405 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
122 feq1 5131 . . . . . 6 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑓:𝑥𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):𝑥𝑆))
123 fveq1 5288 . . . . . . . 8 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑓𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦))
124 reseq1 4695 . . . . . . . . 9 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑓𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))
125124fveq2d 5293 . . . . . . . 8 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝐺‘(𝑓𝑦)) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦)))
126123, 125eqeq12d 2102 . . . . . . 7 (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑦))))
127126ralbidv 2380 . . . . . 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 2381 . . . 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 2760 . . 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 664  w3a 924  wal 1287   = wceq 1289  wcel 1438  {cab 2074  wne 2255  wral 2359  wrex 2360  Vcvv 2619  cun 2995  wss 2997  {csn 3441  cop 3444   cuni 3648  Ord word 4180  Oncon0 4181  suc csuc 4183  dom cdm 4428  cres 4430  Fun wfun 4996   Fn wfn 4997  wf 4998  cfv 5002  recscrecs 6051
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by:  tfrcllembacc  6102  tfrcllemres  6109
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