Theorem pwfseqlem4a 10634

Description: Lemma for pwfseqlem4 10635. (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 9609	. . 3 ⊢ (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3		vex 3461	. . . . . . 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 10632	. . . . . . 7 ⊢ ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
12	2, 3, 11	sylancl 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13		f1of 6810	. . . . . . . . 9 ⊢ (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω⟶𝑋)
14	6, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻:ω⟶𝑋)
15	14, 5	fssd 6713	. . . . . . 7 ⊢ (𝜑 → 𝐻:ω⟶𝐴)
16		ficardom 9935	. . . . . . 7 ⊢ (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17		ffvelcdm 7066	. . . . . . 7 ⊢ ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
18	15, 16, 17	syl2an 607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
19	12, 18	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
20	19	ex 417	. . . 4 ⊢ (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
21	20	adantr 485	. . 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 9603	. . . . 5 ⊢ ω ∈ On
24		onenon 9923	. . . . 5 ⊢ (ω ∈ On → ω ∈ dom card)
25	23, 24	ax-mp 5	. . . 4 ⊢ ω ∈ dom card
26		simpr3 1213	. . . . . 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 10007	. . . . 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 9963	. . . 4 ⊢ ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
31	25, 29, 30	sylancr 598	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
32		nfv 1937	. . . . . . 7 ⊢ Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
33		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑎
34		nfmpo2 7481	. . . . . . . . . 10 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
35	10, 34	nfcxfr 2925	. . . . . . . . 9 ⊢ Ⅎ𝑟𝐹
36		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑠
37	33, 35, 36	nfov 7430	. . . . . . . 8 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
38	37	nfel1 2943	. . . . . . 7 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
39	32, 38	nfim 1919	. . . . . 6 ⊢ Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
40		sseq1 3964	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
41		weeq1 5639	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
42	40, 41	3anbi23d 1463	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
43	42	anbi1d 642	. . . . . . . 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 641	. . . . . . 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 7408	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
46	45	eleq1d 2850	. . . . . . 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 1937	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
49		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑎
50		nfmpo1 7480	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
51	10, 50	nfcxfr 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
52		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑟
53	49, 51, 52	nfov 7430	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
54	53	nfel1 2943	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
55	48, 54	nfim 1919	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
56		sseq1 3964	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
57		xpeq12 5677	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
58	57	anidms 576	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
59	58	sseq2d 3971	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
60		weeq2 5640	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
61	56, 59, 60	3anbi123d 1460	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
62		breq2 5109	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
63	61, 62	anbi12d 643	. . . . . . . . . 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 641	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
66		oveq1 7407	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
67		difeq2 4077	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
68	66, 67	eleq12d 2859	. . . . . . . 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 10633	. . . . . . 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 3919	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
74	73	expr 461	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {crab 3417  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ifcif 4483  𝒫 cpw 4558  ∩ cint 4908  ∪ ciun 4952   class class class wbr 5105   We wwe 5604   × cxp 5650  ^◡ccnv 5651  dom cdm 5652  ran crn 5653  Oncon0 6350  ⟶wf 6521  –1-1→wf1 6522  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ωcom 7850   ↑_m cmap 8812   ≼ cdom 8929   ≺ csdm 8930  Fincfn 8931  cardccrd 9909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-card 9913

This theorem is referenced by:  pwfseqlem4  10635