Theorem pwfseqlem4a 10086

Description: Lemma for pwfseqlem4 10087. (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 9118	. . 3 ⊢ (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3		vex 3500	. . . . . . 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 10084	. . . . . . 7 ⊢ ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
12	2, 3, 11	sylancl 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13		f1of 6618	. . . . . . . . 9 ⊢ (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω⟶𝑋)
14	6, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻:ω⟶𝑋)
15	14, 5	fssd 6531	. . . . . . 7 ⊢ (𝜑 → 𝐻:ω⟶𝐴)
16		ficardom 9393	. . . . . . 7 ⊢ (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17		ffvelrn 6852	. . . . . . 7 ⊢ ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
18	15, 16, 17	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
19	12, 18	eqeltrd 2916	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
20	19	ex 415	. . . 4 ⊢ (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
21	20	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
22	1, 21	syl5bir 245	. 2 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
23		omelon 9112	. . . . 5 ⊢ ω ∈ On
24		onenon 9381	. . . . 5 ⊢ (ω ∈ On → ω ∈ dom card)
25	23, 24	ax-mp 5	. . . 4 ⊢ ω ∈ dom card
26		simpr3 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
27	26	19.8ad 2180	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
28		ween 9464	. . . . 5 ⊢ (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
29	27, 28	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
30		domtri2 9421	. . . 4 ⊢ ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
31	25, 29, 30	sylancr 589	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
32		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
33		nfcv 2980	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑎
34		nfmpo2 7238	. . . . . . . . . 10 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
35	10, 34	nfcxfr 2978	. . . . . . . . 9 ⊢ Ⅎ𝑟𝐹
36		nfcv 2980	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑠
37	33, 35, 36	nfov 7189	. . . . . . . 8 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
38	37	nfel1 2997	. . . . . . 7 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
39	32, 38	nfim 1896	. . . . . 6 ⊢ Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
40		sseq1 3995	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
41		weeq1 5546	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
42	40, 41	3anbi23d 1435	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
43	42	anbi1d 631	. . . . . . . 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 630	. . . . . . 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 7167	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
46	45	eleq1d 2900	. . . . . . 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 1914	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
49		nfcv 2980	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑎
50		nfmpo1 7237	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
51	10, 50	nfcxfr 2978	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
52		nfcv 2980	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑟
53	49, 51, 52	nfov 7189	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
54	53	nfel1 2997	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
55	48, 54	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
56		sseq1 3995	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
57		xpeq12 5583	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
58	57	anidms 569	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
59	58	sseq2d 4002	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
60		weeq2 5547	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
61	56, 59, 60	3anbi123d 1432	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
62		breq2 5073	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
63	61, 62	anbi12d 632	. . . . . . . . . 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	syl5bb 285	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
65	64	anbi2d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
66		oveq1 7166	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
67		difeq2 4096	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
68	66, 67	eleq12d 2910	. . . . . . . 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 10085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
71	55, 69, 70	chvarfv 2241	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
72	39, 47, 71	chvarfv 2241	. . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
73	72	eldifad 3951	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
74	73	expr 459	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → (𝑎𝐹𝑠) ∈ 𝐴))
75	31, 74	sylbird 262	. 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {crab 3145  Vcvv 3497   ∖ cdif 3936   ⊆ wss 3939  ifcif 4470  𝒫 cpw 4542  ∩ cint 4879  ∪ ciun 4922   class class class wbr 5069   We wwe 5516   × cxp 5556  ^◡ccnv 5557  dom cdm 5558  ran crn 5559  Oncon0 6194  ⟶wf 6354  –1-1→wf1 6355  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159   ∈ cmpo 7161  ωcom 7583   ↑_m cmap 8409   ≼ cdom 8510   ≺ csdm 8511  Fincfn 8512  cardccrd 9367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-map 8411  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-card 9371

This theorem is referenced by:  pwfseqlem4  10087