pwfseqlem5 - Metamath Proof Explorer

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Theorem pwfseqlem5 10732

Description: Lemma for pwfseq 10733. Although in some ways pwfseqlem4 10731 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection 𝐾 from the set of finite sequences on an infinite set 𝑥 to 𝑥. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 10096. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 10083. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 9775), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 10087). (Contributed by Mario Carneiro, 31-May-2015.)

**Hypotheses**
Ref	Expression
pwfseqlem5.g	⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
pwfseqlem5.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem5.h	⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem5.ps	⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
pwfseqlem5.n	⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
pwfseqlem5.o	⊢ 𝑂 = OrdIso(𝑟, 𝑡)
pwfseqlem5.t	⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)
pwfseqlem5.p	⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇)
pwfseqlem5.s	⊢ 𝑆 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑_m suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})
pwfseqlem5.q	⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
pwfseqlem5.i	⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)
pwfseqlem5.k	⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)

**Assertion**
Ref	Expression
pwfseqlem5	⊢ ¬ 𝜑

Distinct variable groups: 𝑛,𝑏,𝐺 𝑟,𝑏,𝑡,𝐻 𝑓,𝑘,𝑥,𝑃 𝑓,𝑏,𝑘,𝑢,𝑣,𝑥,𝑦,𝑛,𝑟,𝑡 𝜑,𝑏,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦 𝐾,𝑏,𝑛 𝑁,𝑏 𝜓,𝑘,𝑛,𝑥,𝑦 𝑆,𝑛,𝑦 𝐴,𝑏,𝑛,𝑟,𝑡 𝑂,𝑏,𝑢,𝑣,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑣,𝑢,𝑓) 𝜓(𝑣,𝑢,𝑡,𝑓,𝑟,𝑏) 𝐴(𝑥,𝑦,𝑣,𝑢,𝑓,𝑘) 𝑃(𝑦,𝑣,𝑢,𝑡,𝑛,𝑟,𝑏) 𝑄(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏) 𝑆(𝑥,𝑣,𝑢,𝑡,𝑓,𝑘,𝑟,𝑏) 𝑇(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏) 𝐺(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑟) 𝐻(𝑥,𝑦,𝑣,𝑢,𝑓,𝑘,𝑛) 𝐼(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏) 𝐾(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑟) 𝑁(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟) 𝑂(𝑡,𝑓,𝑘,𝑛,𝑟) 𝑋(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏)

Proof of Theorem pwfseqlem5 Dummy variables 𝑎 𝑐 𝑑 𝑖 𝑗 𝑚 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwfseqlem5.g	. 2 ⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
2		pwfseqlem5.x	. 2 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
3		pwfseqlem5.h	. 2 ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
4		pwfseqlem5.ps	. 2 ⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
5		vex 3492	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
6		simprl3 1220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
7	4, 6	sylan2b 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑟 We 𝑡)
8		pwfseqlem5.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑟, 𝑡)
9	8	oiiso 9606	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
10	5, 7, 9	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11		isof1o 7359	. . . . . . . . . 10 ⊢ (𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡) → 𝑂:dom 𝑂–1-1-onto→𝑡)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂–1-1-onto→𝑡)
13		cardom 10055	. . . . . . . . . . . 12 ⊢ (card‘ω) = ω
14		simprr 772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
15	4, 14	sylan2b 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑡)
16	8	oien 9607	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂 ≈ 𝑡)
17	5, 7, 16	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≈ 𝑡)
18	17	ensymd 9065	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≈ dom 𝑂)
19		domentr 9073	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝑡 ∧ 𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
20	15, 18, 19	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ dom 𝑂)
21		omelon 9715	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
22		onenon 10018	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
24	8	oion 9605	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ V → dom 𝑂 ∈ On)
25	24	elv 3493	. . . . . . . . . . . . . . 15 ⊢ dom 𝑂 ∈ On
26		onenon 10018	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
27	25, 26	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ dom card)
28		carddom2 10046	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
29	23, 27, 28	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
30	20, 29	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
31	13, 30	eqsstrrid 4058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ (card‘dom 𝑂))
32		cardonle 10026	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
33	25, 32	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
34	31, 33	sstrd 4019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ dom 𝑂)
35		sseq2 4035	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑁‘𝑏) = (𝑁‘dom 𝑂))
37	36	f1oeq1d 6857	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏))
38		xpeq12 5725	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = dom 𝑂 ∧ 𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
39	38	anidms 566	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
40	39	f1oeq2d 6858	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑏))
41		f1oeq3 6852	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
42	37, 40, 41	3bitrd 305	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
43	35, 42	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑏 = dom 𝑂 → ((ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)))
44		pwfseqlem5.n	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
45	44	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
46	25	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ On)
47	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
48		omex 9712	. . . . . . . . . . . . . . . . . 18 ⊢ ω ∈ V
49		ovex 7481	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ↑_m 𝑛) ∈ V
50	48, 49	iunex 8009	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
51		f1dmex 7997	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
52	47, 50, 51	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝒫 𝐴 ∈ V)
53		pwexb 7801	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
54	52, 53	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ V)
55		simprl1 1218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡 ⊆ 𝐴)
56	4, 55	sylan2b 593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ⊆ 𝐴)
57		ssdomg 9060	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → (𝑡 ⊆ 𝐴 → 𝑡 ≼ 𝐴))
58	54, 56, 57	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝐴)
59		canth2g 9197	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
60		sdomdom 9040	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
61	54, 59, 60	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≼ 𝒫 𝐴)
62		domtr 9067	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
63	58, 61, 62	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝒫 𝐴)
64		endomtr 9072	. . . . . . . . . . . . 13 ⊢ ((dom 𝑂 ≈ 𝑡 ∧ 𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
65	17, 63, 64	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
66		elharval 9630	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
67	46, 65, 66	sylanbrc 582	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
68	43, 45, 67	rspcdva 3636	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
69	34, 68	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)
70		f1oco 6885	. . . . . . . . 9 ⊢ ((𝑂:dom 𝑂–1-1-onto→𝑡 ∧ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡)
71	12, 69, 70	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡)
72		f1of 6862	. . . . . . . . . . . . . . 15 ⊢ (𝑂:dom 𝑂–1-1-onto→𝑡 → 𝑂:dom 𝑂⟶𝑡)
73	12, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂⟶𝑡)
74	73	feqmptd 6990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)))
75	74	f1oeq1d 6857	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑂:dom 𝑂–1-1-onto→𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂–1-1-onto→𝑡))
76	12, 75	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂–1-1-onto→𝑡)
77	73	feqmptd 6990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)))
78	77	f1oeq1d 6857	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑂:dom 𝑂–1-1-onto→𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂–1-1-onto→𝑡))
79	12, 78	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂–1-1-onto→𝑡)
80	76, 79	xpf1o 9205	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
81		pwfseqlem5.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)
82		f1oeq1 6850	. . . . . . . . . . 11 ⊢ (𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩) → (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡)))
83	81, 82	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
84	80, 83	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
85		f1ocnv 6874	. . . . . . . . 9 ⊢ (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) → ^◡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
86	84, 85	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ^◡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
87		f1oco 6885	. . . . . . . 8 ⊢ (((𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡 ∧ ^◡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂)) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡)
88	71, 86, 87	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡)
89		pwfseqlem5.p	. . . . . . . 8 ⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇)
90		f1oeq1 6850	. . . . . . . 8 ⊢ (𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇) → (𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡))
91	89, 90	ax-mp 5	. . . . . . 7 ⊢ (𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ^◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡)
92	88, 91	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡)
93		f1of1 6861	. . . . . 6 ⊢ (𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡 → 𝑃:(𝑡 × 𝑡)–1-1→𝑡)
94	92, 93	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)–1-1→𝑡)
95		f1of1 6861	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂–1-1-onto→𝑡 → 𝑂:dom 𝑂–1-1→𝑡)
96	12, 95	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂–1-1→𝑡)
97		f1ssres 6824	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂–1-1→𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1→𝑡)
98	96, 34, 97	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω–1-1→𝑡)
99		f1f1orn 6873	. . . . . . . . . . 11 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
100	98, 99	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
101	73, 34	feqresmpt 6991	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)))
102	101	f1oeq1d 6857	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
103	100, 102	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω))
104		mptresid 6080	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡) = (𝑦 ∈ 𝑡 ↦ 𝑦)
105	104	eqcomi 2749	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡)
106		f1oi 6900	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡
107		f1oeq1 6850	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡) → ((𝑦 ∈ 𝑡 ↦ 𝑦):𝑡–1-1-onto→𝑡 ↔ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡))
108	106, 107	mpbiri 258	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡) → (𝑦 ∈ 𝑡 ↦ 𝑦):𝑡–1-1-onto→𝑡)
109	105, 108	mp1i 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑦 ∈ 𝑡 ↦ 𝑦):𝑡–1-1-onto→𝑡)
110	103, 109	xpf1o 9205	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
111		pwfseqlem5.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)
112		f1oeq1 6850	. . . . . . . . 9 ⊢ (𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩) → (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡)))
113	111, 112	ax-mp 5	. . . . . . . 8 ⊢ (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
114	110, 113	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
115		f1of1 6861	. . . . . . 7 ⊢ (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
116	114, 115	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
117		f1f 6817	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
118		frn 6754	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
119		xpss1 5719	. . . . . . 7 ⊢ (ran (𝑂 ↾ ω) ⊆ 𝑡 → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
120	98, 117, 118, 119	4syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
121		f1ss 6822	. . . . . 6 ⊢ ((𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡) ∧ (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡)) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
122	116, 120, 121	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
123		f1co 6828	. . . . 5 ⊢ ((𝑃:(𝑡 × 𝑡)–1-1→𝑡 ∧ 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡)) → (𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡)
124	94, 122, 123	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡)
125	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ∈ V)
126		peano1 7927	. . . . . . . 8 ⊢ ∅ ∈ ω
127	126	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ ω)
128	34, 127	sseldd 4009	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ dom 𝑂)
129	73, 128	ffvelcdmd 7119	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂‘∅) ∈ 𝑡)
130		pwfseqlem5.s	. . . . 5 ⊢ 𝑆 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑_m suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})
131		pwfseqlem5.q	. . . . 5 ⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
132	125, 129, 92, 130, 131	fseqenlem2 10094	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→(ω × 𝑡))
133		f1co 6828	. . . 4 ⊢ (((𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡 ∧ 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→(ω × 𝑡)) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
134	124, 132, 133	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
135		pwfseqlem5.k	. . . 4 ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
136		f1eq1 6812	. . . 4 ⊢ (𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄) → (𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡))
137	135, 136	ax-mp 5	. . 3 ⊢ (𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
138	134, 137	sylibr 234	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
139		eqid 2740	. 2 ⊢ (𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))}) = (𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})
140		eqid 2740	. 2 ⊢ (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡}))) = (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))
141		eqid 2740	. . 3 ⊢ {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
142	141	fpwwe2cbv 10699	. 2 ⊢ {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(^◡𝑠 “ {𝑏}) / 𝑤](𝑤(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑠 ∩ (𝑤 × 𝑤))) = 𝑏))}
143		eqid 2740	. 2 ⊢ ∪ dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = ∪ dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(^◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((^◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (^◡𝐺‘(^◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
144	1, 2, 3, 4, 138, 139, 140, 142, 143	pwfseqlem4 10731	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 [wsbc 3804 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 ifcif 4548 𝒫 cpw 4622 {csn 4648 ⟨cop 4654 ∪ cuni 4931 ∩ cint 4970 ∪ ciun 5015 class class class wbr 5166 {copab 5228 ↦ cmpt 5249 I cid 5592 E cep 5598 We wwe 5651 × cxp 5698 ^◡ccnv 5699 dom cdm 5700 ran crn 5701 ↾ cres 5702 “ cima 5703 ∘ ccom 5704 Oncon0 6395 suc csuc 6397 ⟶wf 6569 –1-1→wf1 6570 –1-1-onto→wf1o 6572 ‘cfv 6573 Isom wiso 6574 (class class class)co 7448 ∈ cmpo 7450 ωcom 7903 seq_ωcseqom 8503 ↑_m cmap 8884 ≈ cen 9000 ≼ cdom 9001 ≺ csdm 9002 Fincfn 9003 OrdIsocoi 9578 harchar 9625 cardccrd 10004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-seqom 8504 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-oi 9579 df-har 9626 df-card 10008

This theorem is referenced by: pwfseq 10733

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