Theorem tfrlemisucaccv 6534

Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6541. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlemisucfn.2	⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
tfrlemisucfn.3	⊢ (𝜑 → 𝑧 ∈ On)
tfrlemisucfn.4	⊢ (𝜑 → 𝑔 Fn 𝑧)
tfrlemisucfn.5	⊢ (𝜑 → 𝑔 ∈ 𝐴)

Assertion

Ref	Expression
tfrlemisucaccv	⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,𝑥,𝑦,𝑧   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑧,𝑓,𝑔)

Proof of Theorem tfrlemisucaccv

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlemisucfn.3	. . . 4 ⊢ (𝜑 → 𝑧 ∈ On)
2		onsuc 4605	. . . 4 ⊢ (𝑧 ∈ On → suc 𝑧 ∈ On)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → suc 𝑧 ∈ On)
4		tfrlemisucfn.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
5		tfrlemisucfn.2	. . . 4 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
6		tfrlemisucfn.4	. . . 4 ⊢ (𝜑 → 𝑔 Fn 𝑧)
7		tfrlemisucfn.5	. . . 4 ⊢ (𝜑 → 𝑔 ∈ 𝐴)
8	4, 5, 1, 6, 7	tfrlemisucfn 6533	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)
9		vex 2806	. . . . . 6 ⊢ 𝑢 ∈ V
10	9	elsuc 4509	. . . . 5 ⊢ (𝑢 ∈ suc 𝑧 ↔ (𝑢 ∈ 𝑧 ∨ 𝑢 = 𝑧))
11		vex 2806	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
12	4, 11	tfrlem3a 6519	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
13	7, 12	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
14		simprrr 542	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
15		simprrl 541	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → 𝑔 Fn 𝑣)
16	6	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → 𝑔 Fn 𝑧)
17		fndmu 5440	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑣 ∧ 𝑔 Fn 𝑧) → 𝑣 = 𝑧)
18	15, 16, 17	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → 𝑣 = 𝑧)
19	18	raleqdv 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → (∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
20	14, 19	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
21	13, 20	rexlimddv 2656	. . . . . . . 8 ⊢ (𝜑 → ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
22	21	r19.21bi 2621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
23		elirrv 4652	. . . . . . . . . . 11 ⊢ ¬ 𝑢 ∈ 𝑢
24		elequ2 2207	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ 𝑢))
25	23, 24	mtbiri 682	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ¬ 𝑢 ∈ 𝑧)
26	25	necon2ai 2457	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑧 → 𝑧 ≠ 𝑢)
27	26	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → 𝑧 ≠ 𝑢)
28		fvunsng 5856	. . . . . . . 8 ⊢ ((𝑢 ∈ V ∧ 𝑧 ≠ 𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝑔‘𝑢))
29	9, 27, 28	sylancr 414	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝑔‘𝑢))
30		eloni 4478	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → Ord 𝑧)
31	1, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑧)
32		ordelss 4482	. . . . . . . . . . 11 ⊢ ((Ord 𝑧 ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ 𝑧)
33	31, 32	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ 𝑧)
34		resabs1 5048	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢))
35	33, 34	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢))
36		elirrv 4652	. . . . . . . . . . . 12 ⊢ ¬ 𝑧 ∈ 𝑧
37		fsnunres 5864	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑧 ∧ ¬ 𝑧 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
38	6, 36, 37	sylancl 413	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
39	38	reseq1d 5018	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔 ↾ 𝑢))
40	39	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔 ↾ 𝑢))
41	35, 40	eqtr3d 2266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = (𝑔 ↾ 𝑢))
42	41	fveq2d 5652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) = (𝐹‘(𝑔 ↾ 𝑢)))
43	22, 29, 42	3eqtr4d 2274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
44	5	tfrlem3-2d 6521	. . . . . . . . . 10 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
45	44	simprd 114	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
46		fndm 5436	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
47	6, 46	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑔 = 𝑧)
48	47	eleq2d 2301	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝑔 ↔ 𝑧 ∈ 𝑧))
49	36, 48	mtbiri 682	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
50		fsnunfv 5863	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ (𝐹‘𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
51	1, 45, 49, 50	syl3anc 1274	. . . . . . . 8 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
52	51	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
53		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → 𝑢 = 𝑧)
54	53	fveq2d 5652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧))
55		reseq2 5014	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧))
56	55, 38	sylan9eqr 2286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = 𝑔)
57	56	fveq2d 5652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) = (𝐹‘𝑔))
58	52, 54, 57	3eqtr4d 2274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
59	43, 58	jaodan 805	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑧 ∨ 𝑢 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
60	10, 59	sylan2b 287	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
61	60	ralrimiva 2606	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
62		fneq2 5426	. . . . 5 ⊢ (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧))
63		raleq 2731	. . . . 5 ⊢ (𝑤 = suc 𝑧 → (∀𝑢 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) ↔ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢))))
64	62, 63	anbi12d 473	. . . 4 ⊢ (𝑤 = suc 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧 ∧ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))))
65	64	rspcev 2911	. . 3 ⊢ ((suc 𝑧 ∈ On ∧ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧 ∧ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))) → ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢))))
66	3, 8, 61, 65	syl12anc 1272	. 2 ⊢ (𝜑 → ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢))))
67		vex 2806	. . . . . 6 ⊢ 𝑧 ∈ V
68		opexg 4326	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
69	67, 45, 68	sylancr 414	. . . . 5 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
70		snexg 4280	. . . . 5 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
71	69, 70	syl 14	. . . 4 ⊢ (𝜑 → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
72		unexg 4546	. . . 4 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
73	11, 71, 72	sylancr 414	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
74	4	tfrlem3ag 6518	. . 3 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))))
75	73, 74	syl 14	. 2 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))))
76	66, 75	mpbird 167	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2202  {cab 2217   ≠ wne 2403  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ∪ cun 3199   ⊆ wss 3201  {csn 3673  ⟨cop 3676  Ord word 4465  Oncon0 4466  suc csuc 4468  dom cdm 4731   ↾ cres 4733  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333

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-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-v 2805  df-sbc 3033  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-br 4094  df-opab 4156  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-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  tfrlemibacc  6535  tfrlemi14d  6542