Theorem pwfseqlem4a 9362

Description: Lemma for pwfseqlem4 9363. (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 8432	. . 3 ⊢ (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3		vex 3176	. . . . . . 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 9360	. . . . . . 7 ⊢ ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
12	2, 3, 11	sylancl 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13		f1of 6050	. . . . . . . . 9 ⊢ (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω⟶𝑋)
14	6, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻:ω⟶𝑋)
15	14, 5	fssd 5970	. . . . . . 7 ⊢ (𝜑 → 𝐻:ω⟶𝐴)
16		ficardom 8670	. . . . . . 7 ⊢ (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17		ffvelrn 6265	. . . . . . 7 ⊢ ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
18	15, 16, 17	syl2an 493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
19	12, 18	eqeltrd 2688	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
20	19	ex 449	. . . 4 ⊢ (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
22	1, 21	syl5bir 232	. 2 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
23		omelon 8426	. . . . 5 ⊢ ω ∈ On
24		onenon 8658	. . . . 5 ⊢ (ω ∈ On → ω ∈ dom card)
25	23, 24	ax-mp 5	. . . 4 ⊢ ω ∈ dom card
26		simpr3 1062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
27		19.8a 2039	. . . . . 6 ⊢ (𝑠 We 𝑎 → ∃𝑠 𝑠 We 𝑎)
28	26, 27	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
29		ween 8741	. . . . 5 ⊢ (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
30	28, 29	sylibr 223	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
31		domtri2 8698	. . . 4 ⊢ ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
32	25, 30, 31	sylancr 694	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
33		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
34		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑎
35		nfmpt22 6621	. . . . . . . . . 10 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
36	10, 35	nfcxfr 2749	. . . . . . . . 9 ⊢ Ⅎ𝑟𝐹
37		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑟𝑠
38	34, 36, 37	nfov 6575	. . . . . . . 8 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
39	38	nfel1 2765	. . . . . . 7 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
40	33, 39	nfim 1813	. . . . . 6 ⊢ Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
41		sseq1 3589	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
42		weeq1 5026	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
43	41, 42	3anbi23d 1394	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
44	43	anbi1d 737	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
45	44	anbi2d 736	. . . . . . 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))))
46		oveq2 6557	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
47	46	eleq1d 2672	. . . . . . 7 ⊢ (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)))
48	45, 47	imbi12d 333	. . . . . 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))))
49		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
50		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑎
51		nfmpt21 6620	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
52	10, 51	nfcxfr 2749	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
53		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑟
54	50, 52, 53	nfov 6575	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
55	54	nfel1 2765	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
56	49, 55	nfim 1813	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
57		sseq1 3589	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
58		xpeq12 5058	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
59	58	anidms 675	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
60	59	sseq2d 3596	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
61		weeq2 5027	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
62	57, 60, 61	3anbi123d 1391	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
63		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
64	62, 63	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
65	7, 64	syl5bb 271	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
66	65	anbi2d 736	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
67		oveq1 6556	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
68		difeq2 3684	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
69	67, 68	eleq12d 2682	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)))
70	66, 69	imbi12d 333	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))))
71	4, 5, 6, 7, 8, 9, 10	pwfseqlem3 9361	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
72	56, 70, 71	chvar 2250	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
73	40, 48, 72	chvar 2250	. . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
74	73	eldifad 3552	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
75	74	expr 641	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → (𝑎𝐹𝑠) ∈ 𝐴))
76	32, 75	sylbird 249	. 2 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
77	22, 76	pm2.61d 169	1 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108  ∩ cint 4410  ∪ ciun 4455   class class class wbr 4583   We wwe 4996   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  Oncon0 5640  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ↑_𝑚 cmap 7744   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  cardccrd 8644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648

This theorem is referenced by:  pwfseqlem4  9363