pwfseqlem5 - Metamath Proof Explorer

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Theorem pwfseqlem5 10658

Description: Lemma for pwfseq 10659. Although in some ways pwfseqlem4 10657 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 10022. 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 10009. 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 9701), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 10013). (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 3479	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
6		simprl3 1221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
7	4, 6	sylan2b 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑟 We 𝑡)
8		pwfseqlem5.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑟, 𝑡)
9	8	oiiso 9532	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
10	5, 7, 9	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11		isof1o 7320	. . . . . . . . . 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 9981	. . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑡)
16	8	oien 9533	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂 ≈ 𝑡)
17	5, 7, 16	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≈ 𝑡)
18	17	ensymd 9001	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≈ dom 𝑂)
19		domentr 9009	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝑡 ∧ 𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
20	15, 18, 19	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ dom 𝑂)
21		omelon 9641	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
22		onenon 9944	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
24	8	oion 9531	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ V → dom 𝑂 ∈ On)
25	24	elv 3481	. . . . . . . . . . . . . . 15 ⊢ dom 𝑂 ∈ On
26		onenon 9944	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
27	25, 26	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ dom card)
28		carddom2 9972	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
29	23, 27, 28	sylancr 588	. . . . . . . . . . . . 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 4032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ (card‘dom 𝑂))
32		cardonle 9952	. . . . . . . . . . . 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 3993	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ dom 𝑂)
35		sseq2 4009	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑁‘𝑏) = (𝑁‘dom 𝑂))
37	36	f1oeq1d 6829	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏))
38		xpeq12 5702	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = dom 𝑂 ∧ 𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
39	38	anidms 568	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
40	39	f1oeq2d 6830	. . . . . . . . . . . . 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 6824	. . . . . . . . . . . . 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 345	. . . . . . . . . . 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 482	. . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
48		omex 9638	. . . . . . . . . . . . . . . . . 18 ⊢ ω ∈ V
49		ovex 7442	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ↑_m 𝑛) ∈ V
50	48, 49	iunex 7955	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
51		f1dmex 7943	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝒫 𝐴 ∈ V)
53		pwexb 7753	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
54	52, 53	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ V)
55		simprl1 1219	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡 ⊆ 𝐴)
56	4, 55	sylan2b 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ⊆ 𝐴)
57		ssdomg 8996	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → (𝑡 ⊆ 𝐴 → 𝑡 ≼ 𝐴))
58	54, 56, 57	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝐴)
59		canth2g 9131	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
60		sdomdom 8976	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
61	54, 59, 60	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≼ 𝒫 𝐴)
62		domtr 9003	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
63	58, 61, 62	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝒫 𝐴)
64		endomtr 9008	. . . . . . . . . . . . 13 ⊢ ((dom 𝑂 ≈ 𝑡 ∧ 𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
65	17, 63, 64	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
66		elharval 9556	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
67	46, 65, 66	sylanbrc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
68	43, 45, 67	rspcdva 3614	. . . . . . . . . 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 6857	. . . . . . . . 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 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡)
72		f1of 6834	. . . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)))
75	74	f1oeq1d 6829	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂–1-1-onto→𝑡)
77	73	feqmptd 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)))
78	77	f1oeq1d 6829	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂–1-1-onto→𝑡)
80	76, 79	xpf1o 9139	. . . . . . . . . 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 6822	. . . . . . . . . . 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 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
85		f1ocnv 6846	. . . . . . . . 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 6857	. . . . . . . 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 585	. . . . . . 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 6822	. . . . . . . 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 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡)
93		f1of1 6833	. . . . . 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 6833	. . . . . . . . . . . . 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 6796	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂–1-1→𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1→𝑡)
98	96, 34, 97	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω–1-1→𝑡)
99		f1f1orn 6845	. . . . . . . . . . 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 6962	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)))
102	101	f1oeq1d 6829	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω))
104		mptresid 6051	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡) = (𝑦 ∈ 𝑡 ↦ 𝑦)
105	104	eqcomi 2742	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡)
106		f1oi 6872	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡
107		f1oeq1 6822	. . . . . . . . . . 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 9139	. . . . . . . 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 6822	. . . . . . . . 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 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
115		f1of1 6833	. . . . . . 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 6788	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
118		frn 6725	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
119		xpss1 5696	. . . . . . 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 6794	. . . . . 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 585	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
123		f1co 6800	. . . . 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 585	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡)
125	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ∈ V)
126		peano1 7879	. . . . . . . 8 ⊢ ∅ ∈ ω
127	126	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ ω)
128	34, 127	sseldd 3984	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ dom 𝑂)
129	73, 128	ffvelcdmd 7088	. . . . 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 10020	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→(ω × 𝑡))
133		f1co 6800	. . . 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 585	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
135		pwfseqlem5.k	. . . 4 ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
136		f1eq1 6783	. . . 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 233	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
139		eqid 2733	. 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 2733	. 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 2733	. . 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 10625	. 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 2733	. 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 10657	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 [wsbc 3778 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ifcif 4529 𝒫 cpw 4603 {csn 4629 ⟨cop 4635 ∪ cuni 4909 ∩ cint 4951 ∪ ciun 4998 class class class wbr 5149 {copab 5211 ↦ cmpt 5232 I cid 5574 E cep 5580 We wwe 5631 × cxp 5675 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 ↾ cres 5679 “ cima 5680 ∘ ccom 5681 Oncon0 6365 suc csuc 6367 ⟶wf 6540 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ‘cfv 6544 Isom wiso 6545 (class class class)co 7409 ∈ cmpo 7411 ωcom 7855 seq_ωcseqom 8447 ↑_m cmap 8820 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 Fincfn 8939 OrdIsocoi 9504 harchar 9551 cardccrd 9930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-seqom 8448 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-oi 9505 df-har 9552 df-card 9934

This theorem is referenced by: pwfseq 10659

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