Theorem tfrcllemaccex 6570

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	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
tfrcllemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfrcllemaccex.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)

Assertion

Ref	Expression
tfrcllemaccex	⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))

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 ℎ 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrcl.x	. . 3 ⊢ (𝜑 → Ord 𝑋)
2		ordelon 4486	. . 3 ⊢ ((Ord 𝑋 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ On)
3	1, 2	sylan 283	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ On)
4		eleq1 2294	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑋 ↔ 𝑤 ∈ 𝑋))
5	4	anbi2d 464	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝜑 ∧ 𝑧 ∈ 𝑋) ↔ (𝜑 ∧ 𝑤 ∈ 𝑋)))
6		feq2 5473	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑔:𝑧⟶𝑆 ↔ 𝑔:𝑤⟶𝑆))
7		raleq 2731	. . . . . 6 ⊢ (𝑧 = 𝑤 → (∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
8	6, 7	anbi12d 473	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
9	8	exbidv 1873	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
10	5, 9	imbi12d 234	. . 3 ⊢ (𝑧 = 𝑤 → (((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
11		eleq1 2294	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝑋 ↔ 𝐶 ∈ 𝑋))
12	11	anbi2d 464	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝜑 ∧ 𝑧 ∈ 𝑋) ↔ (𝜑 ∧ 𝐶 ∈ 𝑋)))
13		feq2 5473	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑔:𝑧⟶𝑆 ↔ 𝑔:𝐶⟶𝑆))
14		raleq 2731	. . . . . 6 ⊢ (𝑧 = 𝐶 → (∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
15	13, 14	anbi12d 473	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
16	15	exbidv 1873	. . . 4 ⊢ (𝑧 = 𝐶 → (∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
17	12, 16	imbi12d 234	. . 3 ⊢ (𝑧 = 𝐶 → (((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
18		tfrcl.f	. . . . . . . . 9 ⊢ 𝐹 = recs(𝐺)
19		tfrcl.g	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐺)
20	19	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → Fun 𝐺)
21	1	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → Ord 𝑋)
22		tfrcl.ex	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
23	22	3expia 1232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
24	23	alrimiv 1922	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
25		feq1 5472	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → (𝑓:𝑥⟶𝑆 ↔ ℎ:𝑥⟶𝑆))
26		fveq2 5648	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (𝐺‘𝑓) = (𝐺‘ℎ))
27	26	eleq1d 2300	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘ℎ) ∈ 𝑆))
28	25, 27	imbi12d 234	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆)))
29	28	cbvalv 1966	. . . . . . . . . . . . . . 15 ⊢ (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀ℎ(ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆))
30	24, 29	sylib 122	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀ℎ(ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆))
31	30	19.21bi 1607	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆))
32	31	3impia 1227	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆)
33	32	3adant1r 1258	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆)
34	33	3adant1r 1258	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆)
35	34	3adant1r 1258	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆)
36		tfrcllemsucfn.1	. . . . . . . . . 10 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
37		fveq1 5647	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (𝑓‘𝑦) = (ℎ‘𝑦))
38		reseq1 5013	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝑦) = (ℎ ↾ 𝑦))
39	38	fveq2d 5652	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(ℎ ↾ 𝑦)))
40	37, 39	eqeq12d 2246	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦))))
41	40	ralbidv 2533	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦))))
42	25, 41	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))))
43	42	rexbidv 2534	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))))
44	43	cbvabv 2357	. . . . . . . . . 10 ⊢ {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {ℎ ∣ ∃𝑥 ∈ 𝑋 (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))}
45	36, 44	eqtri 2252	. . . . . . . . 9 ⊢ 𝐴 = {ℎ ∣ ∃𝑥 ∈ 𝑋 (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))}
46		feq1 5472	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑎 → (𝑟:𝑡⟶𝑆 ↔ 𝑎:𝑡⟶𝑆))
47		eleq1 2294	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑎 → (𝑟 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
48		id 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑎 → 𝑟 = 𝑎)
49		fveq2 5648	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝑎 → (𝐺‘𝑟) = (𝐺‘𝑎))
50	49	opeq2d 3874	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑎 → ⟨𝑡, (𝐺‘𝑟)⟩ = ⟨𝑡, (𝐺‘𝑎)⟩)
51	50	sneqd 3686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑎 → {⟨𝑡, (𝐺‘𝑟)⟩} = {⟨𝑡, (𝐺‘𝑎)⟩})
52	48, 51	uneq12d 3364	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑎 → (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩}) = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩}))
53	52	eqeq2d 2243	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑎 → (𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩})))
54	46, 47, 53	3anbi123d 1349	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑎 → ((𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩})) ↔ (𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩}))))
55	54	cbvexv 1967	. . . . . . . . . . . . 13 ⊢ (∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩})) ↔ ∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩})))
56	55	rexbii 2540	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ 𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩})) ↔ ∃𝑡 ∈ 𝑧 ∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩})))
57		feq2 5473	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑏 → (𝑎:𝑡⟶𝑆 ↔ 𝑎:𝑏⟶𝑆))
58		opeq1 3867	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑏 → ⟨𝑡, (𝐺‘𝑎)⟩ = ⟨𝑏, (𝐺‘𝑎)⟩)
59	58	sneqd 3686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑏 → {⟨𝑡, (𝐺‘𝑎)⟩} = {⟨𝑏, (𝐺‘𝑎)⟩})
60	59	uneq2d 3363	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩}) = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))
61	60	eqeq2d 2243	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑏 → (𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})))
62	57, 61	3anbi13d 1351	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑏 → ((𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩})) ↔ (𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))))
63	62	exbidv 1873	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩})) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))))
64	63	cbvrexv 2769	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ 𝑧 ∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺‘𝑎)⟩})) ↔ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})))
65	56, 64	bitri 184	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ 𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩})) ↔ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})))
66	65	abbii 2347	. . . . . . . . . 10 ⊢ {𝑠 ∣ ∃𝑡 ∈ 𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩}))} = {𝑠 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))}
67		eqeq1 2238	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑑 → (𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}) ↔ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})))
68	67	3anbi3d 1355	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑑 → ((𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})) ↔ (𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))))
69	68	exbidv 1873	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑑 → (∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))))
70	69	rexbidv 2534	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑑 → (∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩})) ↔ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))))
71	70	cbvabv 2357	. . . . . . . . . 10 ⊢ {𝑠 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))} = {𝑑 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))}
72	66, 71	eqtri 2252	. . . . . . . . 9 ⊢ {𝑠 ∣ ∃𝑡 ∈ 𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺‘𝑟)⟩}))} = {𝑑 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺‘𝑎)⟩}))}
73		tfrcllemaccex.u	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
74	73	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
75	74	adantlr 477	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
76	75	adantlr 477	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
77		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
78		simpr 110	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → 𝑏 ∈ 𝑧)
79		simplr 529	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → 𝑧 ∈ 𝑋)
80		ordtr1 4491	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑋 → ((𝑏 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) → 𝑏 ∈ 𝑋))
81	1, 80	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑏 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) → 𝑏 ∈ 𝑋))
82	81	ad4antr 494	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → ((𝑏 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) → 𝑏 ∈ 𝑋))
83	78, 79, 82	mp2and 433	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → 𝑏 ∈ 𝑋)
84		eleq1 2294	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑏 → (𝑤 ∈ 𝑋 ↔ 𝑏 ∈ 𝑋))
85		feq2 5473	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑏 → (𝑔:𝑤⟶𝑆 ↔ 𝑔:𝑏⟶𝑆))
86		raleq 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑏 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
87	85, 86	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑏 → ((𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
88	87	exbidv 1873	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑏 → (∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
89	84, 88	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑏 → ((𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝑏 ∈ 𝑋 → ∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
90		simpllr 536	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
91	89, 90, 78	rspcdva 2916	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → (𝑏 ∈ 𝑋 → ∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
92		feq1 5472	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑎 → (𝑔:𝑏⟶𝑆 ↔ 𝑎:𝑏⟶𝑆))
93		fveq1 5647	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑎 → (𝑔‘𝑢) = (𝑎‘𝑢))
94		reseq1 5013	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑎 → (𝑔 ↾ 𝑢) = (𝑎 ↾ 𝑢))
95	94	fveq2d 5652	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑎 → (𝐺‘(𝑔 ↾ 𝑢)) = (𝐺‘(𝑎 ↾ 𝑢)))
96	93, 95	eqeq12d 2246	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑎 → ((𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))))
97	96	ralbidv 2533	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑎 → (∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))))
98	92, 97	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑎 → ((𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)))))
99	98	cbvexv 1967	. . . . . . . . . . . . 13 ⊢ (∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))))
100		fveq2 5648	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑐 → (𝑎‘𝑢) = (𝑎‘𝑐))
101		reseq2 5014	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑐 → (𝑎 ↾ 𝑢) = (𝑎 ↾ 𝑐))
102	101	fveq2d 5652	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑐 → (𝐺‘(𝑎 ↾ 𝑢)) = (𝐺‘(𝑎 ↾ 𝑐)))
103	100, 102	eqeq12d 2246	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑐 → ((𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)) ↔ (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
104	103	cbvralv 2768	. . . . . . . . . . . . . . 15 ⊢ (∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)) ↔ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))
105	104	anbi2i 457	. . . . . . . . . . . . . 14 ⊢ ((𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))) ↔ (𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
106	105	exbii 1654	. . . . . . . . . . . . 13 ⊢ (∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
107	99, 106	bitri 184	. . . . . . . . . . . 12 ⊢ (∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
108	91, 107	imbitrdi 161	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → (𝑏 ∈ 𝑋 → ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))))
109	83, 108	mpd 13	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
110	109	ralrimiva 2606	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))
111	18, 20, 21, 35, 45, 72, 76, 77, 110	tfrcllemex 6569	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → ∃ℎ(ℎ:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢))))
112		feq1 5472	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → (ℎ:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
113		fveq1 5647	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑔 → (ℎ‘𝑢) = (𝑔‘𝑢))
114		reseq1 5013	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑔 → (ℎ ↾ 𝑢) = (𝑔 ↾ 𝑢))
115	114	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑔 → (𝐺‘(ℎ ↾ 𝑢)) = (𝐺‘(𝑔 ↾ 𝑢)))
116	113, 115	eqeq12d 2246	. . . . . . . . . . 11 ⊢ (ℎ = 𝑔 → ((ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢)) ↔ (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
117	116	ralbidv 2533	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → (∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
118	112, 117	anbi12d 473	. . . . . . . . 9 ⊢ (ℎ = 𝑔 → ((ℎ:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢))) ↔ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
119	118	cbvexv 1967	. . . . . . . 8 ⊢ (∃ℎ(ℎ:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
120	111, 119	sylib 122	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
121	120	exp31 364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
122	121	expcom 116	. . . . 5 ⊢ (𝑧 ∈ On → (𝜑 → (∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))))
123	122	a2d 26	. . . 4 ⊢ (𝑧 ∈ On → ((𝜑 → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) → (𝜑 → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))))
124		impexp 263	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝜑 → (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
125	124	ralbii 2539	. . . . 5 ⊢ (∀𝑤 ∈ 𝑧 ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ ∀𝑤 ∈ 𝑧 (𝜑 → (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
126		r19.21v 2610	. . . . 5 ⊢ (∀𝑤 ∈ 𝑧 (𝜑 → (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ↔ (𝜑 → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
127	125, 126	bitri 184	. . . 4 ⊢ (∀𝑤 ∈ 𝑧 ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝜑 → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
128		impexp 263	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝜑 → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
129	123, 127, 128	3imtr4g 205	. . 3 ⊢ (𝑧 ∈ On → (∀𝑤 ∈ 𝑧 ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))
130	10, 17, 129	tfis3 4690	. 2 ⊢ (𝐶 ∈ On → ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
131	3, 130	mpcom 36	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512   ∪ cun 3199  {csn 3673  ⟨cop 3676  ∪ cuni 3898  Ord word 4465  Oncon0 4466  suc csuc 4468   ↾ cres 4733  Fun wfun 5327  ⟶wf 5329  ‘cfv 5333  recscrecs 6513

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfrcllemres  6571