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Theorem tfrcllemaccex 6188
Description: We can define an acceptable function on any element of 𝑋.

As with many of the transfinite recursion theorems, we have hypotheses that state that 𝐹 is a function and that it is defined up to 𝑋. (Contributed by Jim Kingdon, 26-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)))}
tfrcllemaccex.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
Assertion
Ref Expression
tfrcllemaccex ((𝜑𝐶𝑋) → ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝐶,𝑔,𝑢   𝑔,𝐺,𝑢,𝑦,𝑥   𝑓,𝐺,𝑥,𝑦   𝑆,𝑔,𝑢,𝑦,𝑥   𝑆,𝑓   𝑦,𝑋,𝑥,𝑓   𝜑,𝑦,𝑥,𝑓
Allowed substitution hints:   𝜑(𝑢,𝑔)   𝐴(𝑓,𝑔)   𝐶(𝑥,𝑦,𝑓)   𝐹(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑋(𝑢,𝑔)

Proof of Theorem tfrcllemaccex
Dummy variables 𝑎 𝑏 𝑐 𝑟 𝑠 𝑡 𝑑 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.x . . 3 (𝜑 → Ord 𝑋)
2 ordelon 4243 . . 3 ((Ord 𝑋𝐶𝑋) → 𝐶 ∈ On)
31, 2sylan 279 . 2 ((𝜑𝐶𝑋) → 𝐶 ∈ On)
4 eleq1 2162 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑋𝑤𝑋))
54anbi2d 455 . . . 4 (𝑧 = 𝑤 → ((𝜑𝑧𝑋) ↔ (𝜑𝑤𝑋)))
6 feq2 5192 . . . . . 6 (𝑧 = 𝑤 → (𝑔:𝑧𝑆𝑔:𝑤𝑆))
7 raleq 2584 . . . . . 6 (𝑧 = 𝑤 → (∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
86, 7anbi12d 460 . . . . 5 (𝑧 = 𝑤 → ((𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
98exbidv 1764 . . . 4 (𝑧 = 𝑤 → (∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
105, 9imbi12d 233 . . 3 (𝑧 = 𝑤 → (((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
11 eleq1 2162 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑋𝐶𝑋))
1211anbi2d 455 . . . 4 (𝑧 = 𝐶 → ((𝜑𝑧𝑋) ↔ (𝜑𝐶𝑋)))
13 feq2 5192 . . . . . 6 (𝑧 = 𝐶 → (𝑔:𝑧𝑆𝑔:𝐶𝑆))
14 raleq 2584 . . . . . 6 (𝑧 = 𝐶 → (∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1513, 14anbi12d 460 . . . . 5 (𝑧 = 𝐶 → ((𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1615exbidv 1764 . . . 4 (𝑧 = 𝐶 → (∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1712, 16imbi12d 233 . . 3 (𝑧 = 𝐶 → (((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ((𝜑𝐶𝑋) → ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
18 tfrcl.f . . . . . . . . 9 𝐹 = recs(𝐺)
19 tfrcl.g . . . . . . . . . 10 (𝜑 → Fun 𝐺)
2019ad3antrrr 479 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → Fun 𝐺)
211ad3antrrr 479 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → Ord 𝑋)
22 tfrcl.ex . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
23223expia 1151 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
2423alrimiv 1813 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
25 feq1 5191 . . . . . . . . . . . . . . . . 17 (𝑓 = → (𝑓:𝑥𝑆:𝑥𝑆))
26 fveq2 5353 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝐺𝑓) = (𝐺))
2726eleq1d 2168 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺) ∈ 𝑆))
2825, 27imbi12d 233 . . . . . . . . . . . . . . . 16 (𝑓 = → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (:𝑥𝑆 → (𝐺) ∈ 𝑆)))
2928cbvalv 1854 . . . . . . . . . . . . . . 15 (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ ∀(:𝑥𝑆 → (𝐺) ∈ 𝑆))
3024, 29sylib 121 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ∀(:𝑥𝑆 → (𝐺) ∈ 𝑆))
313019.21bi 1505 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (:𝑥𝑆 → (𝐺) ∈ 𝑆))
32313impia 1146 . . . . . . . . . . . 12 ((𝜑𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
33323adant1r 1177 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ 𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
34333adant1r 1177 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
35343adant1r 1177 . . . . . . . . 9 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
36 tfrcllemsucfn.1 . . . . . . . . . 10 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
37 fveq1 5352 . . . . . . . . . . . . . . 15 (𝑓 = → (𝑓𝑦) = (𝑦))
38 reseq1 4749 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓𝑦) = (𝑦))
3938fveq2d 5357 . . . . . . . . . . . . . . 15 (𝑓 = → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑦)))
4037, 39eqeq12d 2114 . . . . . . . . . . . . . 14 (𝑓 = → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑦) = (𝐺‘(𝑦))))
4140ralbidv 2396 . . . . . . . . . . . . 13 (𝑓 = → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦))))
4225, 41anbi12d 460 . . . . . . . . . . . 12 (𝑓 = → ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))))
4342rexbidv 2397 . . . . . . . . . . 11 (𝑓 = → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))))
4443cbvabv 2223 . . . . . . . . . 10 {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = { ∣ ∃𝑥𝑋 (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))}
4536, 44eqtri 2120 . . . . . . . . 9 𝐴 = { ∣ ∃𝑥𝑋 (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))}
46 feq1 5191 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑟:𝑡𝑆𝑎:𝑡𝑆))
47 eleq1 2162 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑟𝐴𝑎𝐴))
48 id 19 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑎𝑟 = 𝑎)
49 fveq2 5353 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑎 → (𝐺𝑟) = (𝐺𝑎))
5049opeq2d 3659 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑎 → ⟨𝑡, (𝐺𝑟)⟩ = ⟨𝑡, (𝐺𝑎)⟩)
5150sneqd 3487 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑎 → {⟨𝑡, (𝐺𝑟)⟩} = {⟨𝑡, (𝐺𝑎)⟩})
5248, 51uneq12d 3178 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑎 → (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}) = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}))
5352eqeq2d 2111 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
5446, 47, 533anbi123d 1258 . . . . . . . . . . . . . 14 (𝑟 = 𝑎 → ((𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ (𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}))))
5554cbvexv 1855 . . . . . . . . . . . . 13 (∃𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
5655rexbii 2401 . . . . . . . . . . . 12 (∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑡𝑧𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
57 feq2 5192 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (𝑎:𝑡𝑆𝑎:𝑏𝑆))
58 opeq1 3652 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → ⟨𝑡, (𝐺𝑎)⟩ = ⟨𝑏, (𝐺𝑎)⟩)
5958sneqd 3487 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → {⟨𝑡, (𝐺𝑎)⟩} = {⟨𝑏, (𝐺𝑎)⟩})
6059uneq2d 3177 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑏 → (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}) = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))
6160eqeq2d 2111 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6257, 613anbi13d 1260 . . . . . . . . . . . . . 14 (𝑡 = 𝑏 → ((𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ (𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6362exbidv 1764 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (∃𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ ∃𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6463cbvrexv 2613 . . . . . . . . . . . 12 (∃𝑡𝑧𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6556, 64bitri 183 . . . . . . . . . . 11 (∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6665abbii 2215 . . . . . . . . . 10 {𝑠 ∣ ∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}))} = {𝑠 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
67 eqeq1 2106 . . . . . . . . . . . . . 14 (𝑠 = 𝑑 → (𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}) ↔ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
68673anbi3d 1264 . . . . . . . . . . . . 13 (𝑠 = 𝑑 → ((𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ (𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6968exbidv 1764 . . . . . . . . . . . 12 (𝑠 = 𝑑 → (∃𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ ∃𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
7069rexbidv 2397 . . . . . . . . . . 11 (𝑠 = 𝑑 → (∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
7170cbvabv 2223 . . . . . . . . . 10 {𝑠 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))} = {𝑑 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
7266, 71eqtri 2120 . . . . . . . . 9 {𝑠 ∣ ∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}))} = {𝑑 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
73 tfrcllemaccex.u . . . . . . . . . . . 12 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
7473adantlr 464 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
7574adantlr 464 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
7675adantlr 464 . . . . . . . . 9 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
77 simpr 109 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → 𝑧𝑋)
78 simpr 109 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑏𝑧)
79 simplr 500 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑧𝑋)
80 ordtr1 4248 . . . . . . . . . . . . . 14 (Ord 𝑋 → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
811, 80syl 14 . . . . . . . . . . . . 13 (𝜑 → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
8281ad4antr 481 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
8378, 79, 82mp2and 427 . . . . . . . . . . 11 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑏𝑋)
84 eleq1 2162 . . . . . . . . . . . . . 14 (𝑤 = 𝑏 → (𝑤𝑋𝑏𝑋))
85 feq2 5192 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑏 → (𝑔:𝑤𝑆𝑔:𝑏𝑆))
86 raleq 2584 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑏 → (∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
8785, 86anbi12d 460 . . . . . . . . . . . . . . 15 (𝑤 = 𝑏 → ((𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
8887exbidv 1764 . . . . . . . . . . . . . 14 (𝑤 = 𝑏 → (∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
8984, 88imbi12d 233 . . . . . . . . . . . . 13 (𝑤 = 𝑏 → ((𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝑏𝑋 → ∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
90 simpllr 504 . . . . . . . . . . . . 13 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
9189, 90, 78rspcdva 2749 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → (𝑏𝑋 → ∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
92 feq1 5191 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → (𝑔:𝑏𝑆𝑎:𝑏𝑆))
93 fveq1 5352 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → (𝑔𝑢) = (𝑎𝑢))
94 reseq1 4749 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑎 → (𝑔𝑢) = (𝑎𝑢))
9594fveq2d 5357 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → (𝐺‘(𝑔𝑢)) = (𝐺‘(𝑎𝑢)))
9693, 95eqeq12d 2114 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑎 → ((𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
9796ralbidv 2396 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → (∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
9892, 97anbi12d 460 . . . . . . . . . . . . . 14 (𝑔 = 𝑎 → ((𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢)))))
9998cbvexv 1855 . . . . . . . . . . . . 13 (∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
100 fveq2 5353 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑐 → (𝑎𝑢) = (𝑎𝑐))
101 reseq2 4750 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑐 → (𝑎𝑢) = (𝑎𝑐))
102101fveq2d 5357 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑐 → (𝐺‘(𝑎𝑢)) = (𝐺‘(𝑎𝑐)))
103100, 102eqeq12d 2114 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑐 → ((𝑎𝑢) = (𝐺‘(𝑎𝑢)) ↔ (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
104103cbvralv 2612 . . . . . . . . . . . . . . 15 (∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢)) ↔ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))
105104anbi2i 448 . . . . . . . . . . . . . 14 ((𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))) ↔ (𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
106105exbii 1552 . . . . . . . . . . . . 13 (∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))) ↔ ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
10799, 106bitri 183 . . . . . . . . . . . 12 (∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
10891, 107syl6ib 160 . . . . . . . . . . 11 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → (𝑏𝑋 → ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))))
10983, 108mpd 13 . . . . . . . . . 10 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
110109ralrimiva 2464 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∀𝑏𝑧𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
11118, 20, 21, 35, 45, 72, 76, 77, 110tfrcllemex 6187 . . . . . . . 8 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∃(:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))))
112 feq1 5191 . . . . . . . . . 10 ( = 𝑔 → (:𝑧𝑆𝑔:𝑧𝑆))
113 fveq1 5352 . . . . . . . . . . . 12 ( = 𝑔 → (𝑢) = (𝑔𝑢))
114 reseq1 4749 . . . . . . . . . . . . 13 ( = 𝑔 → (𝑢) = (𝑔𝑢))
115114fveq2d 5357 . . . . . . . . . . . 12 ( = 𝑔 → (𝐺‘(𝑢)) = (𝐺‘(𝑔𝑢)))
116113, 115eqeq12d 2114 . . . . . . . . . . 11 ( = 𝑔 → ((𝑢) = (𝐺‘(𝑢)) ↔ (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
117116ralbidv 2396 . . . . . . . . . 10 ( = 𝑔 → (∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
118112, 117anbi12d 460 . . . . . . . . 9 ( = 𝑔 → ((:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))) ↔ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
119118cbvexv 1855 . . . . . . . 8 (∃(:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))) ↔ ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
120111, 119sylib 121 . . . . . . 7 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
121120exp31 359 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑧𝑋 → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
122121expcom 115 . . . . 5 (𝑧 ∈ On → (𝜑 → (∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑧𝑋 → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))))
123122a2d 26 . . . 4 (𝑧 ∈ On → ((𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → (𝜑 → (𝑧𝑋 → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))))
124 impexp 261 . . . . . 6 (((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
125124ralbii 2400 . . . . 5 (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ∀𝑤𝑧 (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
126 r19.21v 2468 . . . . 5 (∀𝑤𝑧 (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ↔ (𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
127125, 126bitri 183 . . . 4 (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
128 impexp 261 . . . 4 (((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → (𝑧𝑋 → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
129123, 127, 1283imtr4g 204 . . 3 (𝑧 ∈ On → (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
13010, 17, 129tfis3 4438 . 2 (𝐶 ∈ On → ((𝜑𝐶𝑋) → ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1313, 130mpcom 36 1 ((𝜑𝐶𝑋) → ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 930  wal 1297   = wceq 1299  wex 1436  wcel 1448  {cab 2086  wral 2375  wrex 2376  cun 3019  {csn 3474  cop 3477   cuni 3683  Ord word 4222  Oncon0 4223  suc csuc 4225  cres 4479  Fun wfun 5053  wf 5055  cfv 5059  recscrecs 6131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-recs 6132
This theorem is referenced by:  tfrcllemres  6189
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