Theorem tfrlemisucaccv 6320

Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6327. (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 4497	. . . 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 6319	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)
9		vex 2740	. . . . . 6 ⊢ 𝑢 ∈ V
10	9	elsuc 4403	. . . . 5 ⊢ (𝑢 ∈ suc 𝑧 ↔ (𝑢 ∈ 𝑧 ∨ 𝑢 = 𝑧))
11		vex 2740	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
12	4, 11	tfrlem3a 6305	. . . . . . . . . 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 540	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))) → ∀𝑢 ∈ 𝑣 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
15		simprrl 539	. . . . . . . . . . . 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 5313	. . . . . . . . . . . 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 2678	. . . . . . . . . 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 2599	. . . . . . . 8 ⊢ (𝜑 → ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
22	21	r19.21bi 2565	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))
23		elirrv 4544	. . . . . . . . . . 11 ⊢ ¬ 𝑢 ∈ 𝑢
24		elequ2 2153	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ 𝑢))
25	23, 24	mtbiri 675	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ¬ 𝑢 ∈ 𝑧)
26	25	necon2ai 2401	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑧 → 𝑧 ≠ 𝑢)
27	26	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → 𝑧 ≠ 𝑢)
28		fvunsng 5706	. . . . . . . 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 4372	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → Ord 𝑧)
31	1, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑧)
32		ordelss 4376	. . . . . . . . . . 11 ⊢ ((Ord 𝑧 ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ 𝑧)
33	31, 32	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ 𝑧)
34		resabs1 4932	. . . . . . . . . 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 4544	. . . . . . . . . . . 12 ⊢ ¬ 𝑧 ∈ 𝑧
37		fsnunres 5714	. . . . . . . . . . . 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 4902	. . . . . . . . . 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 2212	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = (𝑔 ↾ 𝑢))
42	41	fveq2d 5515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) = (𝐹‘(𝑔 ↾ 𝑢)))
43	22, 29, 42	3eqtr4d 2220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
44	5	tfrlem3-2d 6307	. . . . . . . . . 10 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
45	44	simprd 114	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
46		fndm 5311	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
47	6, 46	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑔 = 𝑧)
48	47	eleq2d 2247	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝑔 ↔ 𝑧 ∈ 𝑧))
49	36, 48	mtbiri 675	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
50		fsnunfv 5713	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ (𝐹‘𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧) = (𝐹‘𝑔))
51	1, 45, 49, 50	syl3anc 1238	. . . . . . . 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 5515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑧))
55		reseq2 4898	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑧))
56	55, 38	sylan9eqr 2232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢) = 𝑔)
57	56	fveq2d 5515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)) = (𝐹‘𝑔))
58	52, 54, 57	3eqtr4d 2220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
59	43, 58	jaodan 797	. . . . 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 2550	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ↾ 𝑢)))
62		fneq2 5301	. . . . 5 ⊢ (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧))
63		raleq 2672	. . . . 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 2841	. . 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 1236	. 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 2740	. . . . . 6 ⊢ 𝑧 ∈ V
68		opexg 4225	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
69	67, 45, 68	sylancr 414	. . . . 5 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
70		snexg 4181	. . . . 5 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
71	69, 70	syl 14	. . . 4 ⊢ (𝜑 → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
72		unexg 4440	. . . 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 6304	. . 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 708  ∀wal 1351   = wceq 1353   ∈ wcel 2148  {cab 2163   ≠ wne 2347  ∀wral 2455  ∃wrex 2456  Vcvv 2737   ∪ cun 3127   ⊆ wss 3129  {csn 3591  ⟨cop 3594  Ord word 4359  Oncon0 4360  suc csuc 4362  dom cdm 4623   ↾ cres 4625  Fun wfun 5206   Fn wfn 5207  ‘cfv 5212

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-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

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-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220

This theorem is referenced by:  tfrlemibacc  6321  tfrlemi14d  6328