Theorem tfrlemisucaccv 6293

Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6300. (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		suceloni 4478	. . . 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 6292	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)
9		vex 2729	. . . . . 6 ⊢ 𝑢 ∈ V
10	9	elsuc 4384	. . . . 5 ⊢ (𝑢 ∈ suc 𝑧 ↔ (𝑢 ∈ 𝑧 ∨ 𝑢 = 𝑧))
11		vex 2729	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
12	4, 11	tfrlem3a 6278	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
13	7, 12	sylib 121	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
14		simprrr 530	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
15		simprrl 529	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → 𝑔 Fn 𝑣)
16	6	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → 𝑔 Fn 𝑧)
17		fndmu 5289	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑣 ∧ 𝑔 Fn 𝑧) → 𝑣 = 𝑧)
18	15, 16, 17	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → 𝑣 = 𝑧)
19	18	raleqdv 2667	. . . . . . . . . 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 146	. . . . . . . . 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 2588	. . . . . . . 8 ⊢ (𝜑 → ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
22	21	r19.21bi 2554	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
23		elirrv 4525	. . . . . . . . . . 11 ⊢ ¬ 𝑢 ∈ 𝑢
24		elequ2 2141	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ 𝑢))
25	23, 24	mtbiri 665	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ¬ 𝑢 ∈ 𝑧)
26	25	necon2ai 2390	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑧 → 𝑧 ≠ 𝑢)
27	26	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → 𝑧 ≠ 𝑢)
28		fvunsng 5679	. . . . . . . 8 ⊢ ((𝑢 ∈ V ∧ 𝑧 ≠ 𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝑔‘𝑢))
29	9, 27, 28	sylancr 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝑔‘𝑢))
30		eloni 4353	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → Ord 𝑧)
31	1, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑧)
32		ordelss 4357	. . . . . . . . . . 11 ⊢ ((Ord 𝑧 ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ 𝑧)
33	31, 32	sylan 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ 𝑧)
34		resabs1 4913	. . . . . . . . . 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 4525	. . . . . . . . . . . 12 ⊢ ¬ 𝑧 ∈ 𝑧
37		fsnunres 5687	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑧 ∧ ¬ 𝑧 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
38	6, 36, 37	sylancl 410	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
39	38	reseq1d 4883	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔 ↾ 𝑢))
40	39	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔 ↾ 𝑢))
41	35, 40	eqtr3d 2200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = (𝑔 ↾ 𝑢))
42	41	fveq2d 5490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) = (𝐹‘(𝑔 ↾ 𝑢)))
43	22, 29, 42	3eqtr4d 2208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
44	5	tfrlem3-2d 6280	. . . . . . . . . 10 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
45	44	simprd 113	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
46		fndm 5287	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
47	6, 46	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑔 = 𝑧)
48	47	eleq2d 2236	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝑔 ↔ 𝑧 ∈ 𝑧))
49	36, 48	mtbiri 665	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
50		fsnunfv 5686	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ (𝐹‘𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
51	1, 45, 49, 50	syl3anc 1228	. . . . . . . 8 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
52	51	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
53		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → 𝑢 = 𝑧)
54	53	fveq2d 5490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧))
55		reseq2 4879	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧))
56	55, 38	sylan9eqr 2221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = 𝑔)
57	56	fveq2d 5490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) = (𝐹‘𝑔))
58	52, 54, 57	3eqtr4d 2208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
59	43, 58	jaodan 787	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑧 ∨ 𝑢 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
60	10, 59	sylan2b 285	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
61	60	ralrimiva 2539	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
62		fneq2 5277	. . . . 5 ⊢ (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧))
63		raleq 2661	. . . . 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 465	. . . 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 2830	. . 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 1226	. 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 2729	. . . . . 6 ⊢ 𝑧 ∈ V
68		opexg 4206	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
69	67, 45, 68	sylancr 411	. . . . 5 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
70		snexg 4163	. . . . 5 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
71	69, 70	syl 14	. . . 4 ⊢ (𝜑 → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
72		unexg 4421	. . . 4 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
73	11, 71, 72	sylancr 411	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
74	4	tfrlem3ag 6277	. . 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 166	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151   ≠ wne 2336  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ∪ cun 3114   ⊆ wss 3116  {csn 3576  ⟨cop 3579  Ord word 4340  Oncon0 4341  suc csuc 4343  dom cdm 4604   ↾ cres 4606  Fun wfun 5182   Fn wfn 5183  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  tfrlemibacc  6294  tfrlemi14d  6301