Theorem pwfseqlem4a 10673

Description: Lemma for pwfseqlem4 10674. (Contributed by Mario Carneiro, 7-Jun-2016.)

Hypotheses

Ref	Expression
pwfseqlem4.g	⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
pwfseqlem4.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem4.h	⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem4.ps	⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k	⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
pwfseqlem4.d	⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
pwfseqlem4.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))

Assertion

Ref	Expression
pwfseqlem4a	⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)

Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑠,𝑎,𝐹   𝑤,𝐺   𝑤,𝐾   𝑟,𝑎,𝑥,𝑧,𝐻,𝑠   𝑛,𝑎,𝜑,𝑠,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑎,𝑛,𝑟,𝑠,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥, 𝑤, 𝑠, 𝑟, 𝑎)   𝐴(𝑤)   𝐷(𝑥, 𝑤, 𝑠, 𝑟, 𝑎)   𝐹(𝑥, 𝑧, 𝑤, 𝑛, 𝑟)   𝐺(𝑥, 𝑧, 𝑛, 𝑠, 𝑟, 𝑎)   𝐻(𝑤, 𝑛)   𝐾(𝑥, 𝑧, 𝑛, 𝑠, 𝑟, 𝑎)   𝑋(𝑥, 𝑧, 𝑤, 𝑛, 𝑠, 𝑟, 𝑎)

Proof of Theorem pwfseqlem4a

Step	Hyp	Ref	Expression
1		isfinite 9634	. . 3 ⊢ (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3		vex 3457	. . . . . . 7 ⊢ 𝑠 ∈ V
4		pwfseqlem4.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
5		pwfseqlem4.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
6		pwfseqlem4.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
7		pwfseqlem4.ps	. . . . . . . 8 ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
8		pwfseqlem4.k	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
9		pwfseqlem4.d	. . . . . . . 8 ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
10		pwfseqlem4.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
11	4, 5, 6, 7, 8, 9, 10	pwfseqlem2 10671	. . . . . . 7 ⊢ ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
12	2, 3, 11	sylancl 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13		f1of 6821	. . . . . . . . 9 ⊢ (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω⟶𝑋)
14	6, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻:ω⟶𝑋)
15	14, 5	fssd 6724	. . . . . . 7 ⊢ (𝜑 → 𝐻:ω⟶𝐴)
16		ficardom 9969	. . . . . . 7 ⊢ (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17		ffvelcdm 7077	. . . . . . 7 ⊢ ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
18	15, 16, 17	syl2an 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
19	12, 18	eqeltrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
20	19	ex 418	. . . 4 ⊢ (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
21	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
22	1, 21	biimtrrid 246	. 2 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
23		omelon 9628	. . . . 5 ⊢ ω ∈ On
24		onenon 9957	. . . . 5 ⊢ (ω ∈ On → ω ∈ dom card)
25	23, 24	ax-mp 5	. . . 4 ⊢ ω ∈ dom card
26		simpr3 1215	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
27	26	19.8ad 2220	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
28		ween 10041	. . . . 5 ⊢ (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
29	27, 28	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
30		domtri2 9997	. . . 4 ⊢ ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
31	25, 29, 30	sylancr 599	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
32		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
33		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑎
34		nfmpo2 7497	. . . . . . . . . 10 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
35	10, 34	nfcxfr 2922	. . . . . . . . 9 ⊢ Ⅎ𝑟𝐹
36		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑠
37	33, 35, 36	nfov 7446	. . . . . . . 8 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
38	37	nfel1 2940	. . . . . . 7 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
39	32, 38	nfim 1929	. . . . . 6 ⊢ Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
40		sseq1 3959	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
41		weeq1 5646	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
42	40, 41	3anbi23d 1467	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
43	42	anbi1d 643	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
44	43	anbi2d 642	. . . . . . 7 ⊢ (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
45		oveq2 7424	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
46	45	eleq1d 2847	. . . . . . 7 ⊢ (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)))
47	44, 46	imbi12d 347	. . . . . 6 ⊢ (𝑟 = 𝑠 → (((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))))
48		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
49		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑎
50		nfmpo1 7496	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
51	10, 50	nfcxfr 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
52		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑟
53	49, 51, 52	nfov 7446	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
54	53	nfel1 2940	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
55	48, 54	nfim 1929	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
56		sseq1 3959	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
57		xpeq12 5684	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
58	57	anidms 577	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
59	58	sseq2d 3966	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
60		weeq2 5647	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
61	56, 59, 60	3anbi123d 1464	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
62		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
63	61, 62	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
64	7, 63	bitrid 286	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
65	64	anbi2d 642	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
66		oveq1 7423	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
67		difeq2 4071	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
68	66, 67	eleq12d 2856	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)))
69	65, 68	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))))
70	4, 5, 6, 7, 8, 9, 10	pwfseqlem3 10672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
71	55, 69, 70	chvarfv 2278	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
72	39, 47, 71	chvarfv 2278	. . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
73	72	eldifad 3914	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
74	73	expr 462	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → (𝑎𝐹𝑠) ∈ 𝐴))
75	31, 74	sylbird 263	. 2 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
76	22, 75	pm2.61d 181	1 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ifcif 4485  𝒫 cpw 4560  ∩ cint 4910  ∪ ciun 4954   class class class wbr 5107   We wwe 5611   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660  Oncon0 6361  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  ωcom 7865   ↑_m cmap 8829   ≼ cdom 8953   ≺ csdm 8954  Fincfn 8955  cardccrd 9943

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9947

This theorem is used by:  pwfseqlem4  10674