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Theorem tfrcllemaccex 6364
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 4385 . . 3 ((Ord 𝑋𝐶𝑋) → 𝐶 ∈ On)
31, 2sylan 283 . 2 ((𝜑𝐶𝑋) → 𝐶 ∈ On)
4 eleq1 2240 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑋𝑤𝑋))
54anbi2d 464 . . . 4 (𝑧 = 𝑤 → ((𝜑𝑧𝑋) ↔ (𝜑𝑤𝑋)))
6 feq2 5351 . . . . . 6 (𝑧 = 𝑤 → (𝑔:𝑧𝑆𝑔:𝑤𝑆))
7 raleq 2673 . . . . . 6 (𝑧 = 𝑤 → (∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
86, 7anbi12d 473 . . . . 5 (𝑧 = 𝑤 → ((𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
98exbidv 1825 . . . 4 (𝑧 = 𝑤 → (∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
105, 9imbi12d 234 . . 3 (𝑧 = 𝑤 → (((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
11 eleq1 2240 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑋𝐶𝑋))
1211anbi2d 464 . . . 4 (𝑧 = 𝐶 → ((𝜑𝑧𝑋) ↔ (𝜑𝐶𝑋)))
13 feq2 5351 . . . . . 6 (𝑧 = 𝐶 → (𝑔:𝑧𝑆𝑔:𝐶𝑆))
14 raleq 2673 . . . . . 6 (𝑧 = 𝐶 → (∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1513, 14anbi12d 473 . . . . 5 (𝑧 = 𝐶 → ((𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1615exbidv 1825 . . . 4 (𝑧 = 𝐶 → (∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1712, 16imbi12d 234 . . 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 492 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → Fun 𝐺)
211ad3antrrr 492 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → Ord 𝑋)
22 tfrcl.ex . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
23223expia 1205 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
2423alrimiv 1874 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → ∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆))
25 feq1 5350 . . . . . . . . . . . . . . . . 17 (𝑓 = → (𝑓:𝑥𝑆:𝑥𝑆))
26 fveq2 5517 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝐺𝑓) = (𝐺))
2726eleq1d 2246 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝐺𝑓) ∈ 𝑆 ↔ (𝐺) ∈ 𝑆))
2825, 27imbi12d 234 . . . . . . . . . . . . . . . 16 (𝑓 = → ((𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ (:𝑥𝑆 → (𝐺) ∈ 𝑆)))
2928cbvalv 1917 . . . . . . . . . . . . . . 15 (∀𝑓(𝑓:𝑥𝑆 → (𝐺𝑓) ∈ 𝑆) ↔ ∀(:𝑥𝑆 → (𝐺) ∈ 𝑆))
3024, 29sylib 122 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ∀(:𝑥𝑆 → (𝐺) ∈ 𝑆))
313019.21bi 1558 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (:𝑥𝑆 → (𝐺) ∈ 𝑆))
32313impia 1200 . . . . . . . . . . . 12 ((𝜑𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
33323adant1r 1231 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ 𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
34333adant1r 1231 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑥𝑋:𝑥𝑆) → (𝐺) ∈ 𝑆)
35343adant1r 1231 . . . . . . . . 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 5516 . . . . . . . . . . . . . . 15 (𝑓 = → (𝑓𝑦) = (𝑦))
38 reseq1 4903 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓𝑦) = (𝑦))
3938fveq2d 5521 . . . . . . . . . . . . . . 15 (𝑓 = → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑦)))
4037, 39eqeq12d 2192 . . . . . . . . . . . . . 14 (𝑓 = → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑦) = (𝐺‘(𝑦))))
4140ralbidv 2477 . . . . . . . . . . . . 13 (𝑓 = → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦))))
4225, 41anbi12d 473 . . . . . . . . . . . 12 (𝑓 = → ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))))
4342rexbidv 2478 . . . . . . . . . . 11 (𝑓 = → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))))
4443cbvabv 2302 . . . . . . . . . 10 {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = { ∣ ∃𝑥𝑋 (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))}
4536, 44eqtri 2198 . . . . . . . . 9 𝐴 = { ∣ ∃𝑥𝑋 (:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))}
46 feq1 5350 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑟:𝑡𝑆𝑎:𝑡𝑆))
47 eleq1 2240 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑟𝐴𝑎𝐴))
48 id 19 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑎𝑟 = 𝑎)
49 fveq2 5517 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑎 → (𝐺𝑟) = (𝐺𝑎))
5049opeq2d 3787 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑎 → ⟨𝑡, (𝐺𝑟)⟩ = ⟨𝑡, (𝐺𝑎)⟩)
5150sneqd 3607 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑎 → {⟨𝑡, (𝐺𝑟)⟩} = {⟨𝑡, (𝐺𝑎)⟩})
5248, 51uneq12d 3292 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑎 → (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}) = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}))
5352eqeq2d 2189 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
5446, 47, 533anbi123d 1312 . . . . . . . . . . . . . 14 (𝑟 = 𝑎 → ((𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ (𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}))))
5554cbvexv 1918 . . . . . . . . . . . . 13 (∃𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
5655rexbii 2484 . . . . . . . . . . . 12 (∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑡𝑧𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
57 feq2 5351 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (𝑎:𝑡𝑆𝑎:𝑏𝑆))
58 opeq1 3780 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → ⟨𝑡, (𝐺𝑎)⟩ = ⟨𝑏, (𝐺𝑎)⟩)
5958sneqd 3607 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → {⟨𝑡, (𝐺𝑎)⟩} = {⟨𝑏, (𝐺𝑎)⟩})
6059uneq2d 3291 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑏 → (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}) = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))
6160eqeq2d 2189 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6257, 613anbi13d 1314 . . . . . . . . . . . . . 14 (𝑡 = 𝑏 → ((𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ (𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6362exbidv 1825 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (∃𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ ∃𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6463cbvrexv 2706 . . . . . . . . . . . 12 (∃𝑡𝑧𝑎(𝑎:𝑡𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6556, 64bitri 184 . . . . . . . . . . 11 (∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6665abbii 2293 . . . . . . . . . 10 {𝑠 ∣ ∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}))} = {𝑠 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
67 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑠 = 𝑑 → (𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}) ↔ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
68673anbi3d 1318 . . . . . . . . . . . . 13 (𝑠 = 𝑑 → ((𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ (𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6968exbidv 1825 . . . . . . . . . . . 12 (𝑠 = 𝑑 → (∃𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ ∃𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
7069rexbidv 2478 . . . . . . . . . . 11 (𝑠 = 𝑑 → (∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
7170cbvabv 2302 . . . . . . . . . 10 {𝑠 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))} = {𝑑 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
7266, 71eqtri 2198 . . . . . . . . 9 {𝑠 ∣ ∃𝑡𝑧𝑟(𝑟:𝑡𝑆𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}))} = {𝑑 ∣ ∃𝑏𝑧𝑎(𝑎:𝑏𝑆𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
73 tfrcllemaccex.u . . . . . . . . . . . 12 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
7473adantlr 477 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
7574adantlr 477 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
7675adantlr 477 . . . . . . . . 9 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
77 simpr 110 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → 𝑧𝑋)
78 simpr 110 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑏𝑧)
79 simplr 528 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑧𝑋)
80 ordtr1 4390 . . . . . . . . . . . . . 14 (Ord 𝑋 → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
811, 80syl 14 . . . . . . . . . . . . 13 (𝜑 → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
8281ad4antr 494 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
8378, 79, 82mp2and 433 . . . . . . . . . . 11 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑏𝑋)
84 eleq1 2240 . . . . . . . . . . . . . 14 (𝑤 = 𝑏 → (𝑤𝑋𝑏𝑋))
85 feq2 5351 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑏 → (𝑔:𝑤𝑆𝑔:𝑏𝑆))
86 raleq 2673 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑏 → (∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
8785, 86anbi12d 473 . . . . . . . . . . . . . . 15 (𝑤 = 𝑏 → ((𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
8887exbidv 1825 . . . . . . . . . . . . . 14 (𝑤 = 𝑏 → (∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
8984, 88imbi12d 234 . . . . . . . . . . . . 13 (𝑤 = 𝑏 → ((𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝑏𝑋 → ∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
90 simpllr 534 . . . . . . . . . . . . 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 2848 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → (𝑏𝑋 → ∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
92 feq1 5350 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → (𝑔:𝑏𝑆𝑎:𝑏𝑆))
93 fveq1 5516 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → (𝑔𝑢) = (𝑎𝑢))
94 reseq1 4903 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑎 → (𝑔𝑢) = (𝑎𝑢))
9594fveq2d 5521 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → (𝐺‘(𝑔𝑢)) = (𝐺‘(𝑎𝑢)))
9693, 95eqeq12d 2192 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑎 → ((𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
9796ralbidv 2477 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → (∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
9892, 97anbi12d 473 . . . . . . . . . . . . . 14 (𝑔 = 𝑎 → ((𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢)))))
9998cbvexv 1918 . . . . . . . . . . . . 13 (∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
100 fveq2 5517 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑐 → (𝑎𝑢) = (𝑎𝑐))
101 reseq2 4904 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑐 → (𝑎𝑢) = (𝑎𝑐))
102101fveq2d 5521 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑐 → (𝐺‘(𝑎𝑢)) = (𝐺‘(𝑎𝑐)))
103100, 102eqeq12d 2192 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑐 → ((𝑎𝑢) = (𝐺‘(𝑎𝑢)) ↔ (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
104103cbvralv 2705 . . . . . . . . . . . . . . 15 (∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢)) ↔ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))
105104anbi2i 457 . . . . . . . . . . . . . 14 ((𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))) ↔ (𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
106105exbii 1605 . . . . . . . . . . . . 13 (∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))) ↔ ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
10799, 106bitri 184 . . . . . . . . . . . 12 (∃𝑔(𝑔:𝑏𝑆 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑎(𝑎:𝑏𝑆 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
10891, 107imbitrdi 161 . . . . . . . . . . 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 2550 . . . . . . . . 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 6363 . . . . . . . 8 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∃(:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))))
112 feq1 5350 . . . . . . . . . 10 ( = 𝑔 → (:𝑧𝑆𝑔:𝑧𝑆))
113 fveq1 5516 . . . . . . . . . . . 12 ( = 𝑔 → (𝑢) = (𝑔𝑢))
114 reseq1 4903 . . . . . . . . . . . . 13 ( = 𝑔 → (𝑢) = (𝑔𝑢))
115114fveq2d 5521 . . . . . . . . . . . 12 ( = 𝑔 → (𝐺‘(𝑢)) = (𝐺‘(𝑔𝑢)))
116113, 115eqeq12d 2192 . . . . . . . . . . 11 ( = 𝑔 → ((𝑢) = (𝐺‘(𝑢)) ↔ (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
117116ralbidv 2477 . . . . . . . . . 10 ( = 𝑔 → (∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
118112, 117anbi12d 473 . . . . . . . . 9 ( = 𝑔 → ((:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))) ↔ (𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
119118cbvexv 1918 . . . . . . . 8 (∃(:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))) ↔ ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
120111, 119sylib 122 . . . . . . 7 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
121120exp31 364 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑧𝑋 → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
122121expcom 116 . . . . 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 263 . . . . . 6 (((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
125124ralbii 2483 . . . . 5 (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ∀𝑤𝑧 (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
126 r19.21v 2554 . . . . 5 (∀𝑤𝑧 (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ↔ (𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
127125, 126bitri 184 . . . 4 (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
128 impexp 263 . . . 4 (((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → (𝑧𝑋 → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
129123, 127, 1283imtr4g 205 . . 3 (𝑧 ∈ On → (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔:𝑤𝑆 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ((𝜑𝑧𝑋) → ∃𝑔(𝑔:𝑧𝑆 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
13010, 17, 129tfis3 4587 . 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 104  w3a 978  wal 1351   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  cun 3129  {csn 3594  cop 3597   cuni 3811  Ord word 4364  Oncon0 4365  suc csuc 4367  cres 4630  Fun wfun 5212  wf 5214  cfv 5218  recscrecs 6307
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-in1 614  ax-in2 615  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-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308
This theorem is referenced by:  tfrcllemres  6365
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