pwfseqlem5 - Metamath Proof Explorer

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Theorem pwfseqlem5 10524

Description: Lemma for pwfseq 10525. Although in some ways pwfseqlem4 10523 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 9888. 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 9875. 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 9567), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 9879). (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 3446	. . . . . . . . . . 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 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑟 We 𝑡)
8		pwfseqlem5.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑟, 𝑡)
9	8	oiiso 9398	. . . . . . . . . . 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 7254	. . . . . . . . . 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 9847	. . . . . . . . . . . 12 ⊢ (card‘ω) = ω
14		simprr 771	. . . . . . . . . . . . . . 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 9399	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂 ≈ 𝑡)
17	5, 7, 16	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≈ 𝑡)
18	17	ensymd 8870	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≈ dom 𝑂)
19		domentr 8878	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝑡 ∧ 𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
20	15, 18, 19	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ dom 𝑂)
21		omelon 9507	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
22		onenon 9810	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
24	8	oion 9397	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ V → dom 𝑂 ∈ On)
25	24	elv 3448	. . . . . . . . . . . . . . 15 ⊢ dom 𝑂 ∈ On
26		onenon 9810	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
27	25, 26	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ dom card)
28		carddom2 9838	. . . . . . . . . . . . . 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 3984	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ (card‘dom 𝑂))
32		cardonle 9818	. . . . . . . . . . . 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 3945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ dom 𝑂)
35		sseq2 3961	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36		fveq2 6829	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑁‘𝑏) = (𝑁‘dom 𝑂))
37	36	f1oeq1d 6766	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏))
38		xpeq12 5649	. . . . . . . . . . . . . . 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 6767	. . . . . . . . . . . . 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 6761	. . . . . . . . . . . . 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 9504	. . . . . . . . . . . . . . . . . 18 ⊢ ω ∈ V
49		ovex 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ↑_m 𝑛) ∈ V
50	48, 49	iunex 7883	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
51		f1dmex 7871	. . . . . . . . . . . . . . . . 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 7682	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
54	52, 53	sylibr 233	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ⊆ 𝐴)
57		ssdomg 8865	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → (𝑡 ⊆ 𝐴 → 𝑡 ≼ 𝐴))
58	54, 56, 57	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝐴)
59		canth2g 9000	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
60		sdomdom 8845	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
61	54, 59, 60	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≼ 𝒫 𝐴)
62		domtr 8872	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
63	58, 61, 62	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝒫 𝐴)
64		endomtr 8877	. . . . . . . . . . . . 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 9422	. . . . . . . . . . . 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 3574	. . . . . . . . . 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 6794	. . . . . . . . 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 6771	. . . . . . . . . . . . . . 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 6897	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)))
75	74	f1oeq1d 6766	. . . . . . . . . . . 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 6897	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)))
78	77	f1oeq1d 6766	. . . . . . . . . . . 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 9008	. . . . . . . . . 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 6759	. . . . . . . . . . 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 6783	. . . . . . . . 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 6794	. . . . . . . 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 6759	. . . . . . . 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 6770	. . . . . 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 6770	. . . . . . . . . . . . 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 6733	. . . . . . . . . . . 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 6782	. . . . . . . . . . 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 6898	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)))
102	101	f1oeq1d 6766	. . . . . . . . . 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 5994	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡) = (𝑦 ∈ 𝑡 ↦ 𝑦)
105	104	eqcomi 2746	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡)
106		f1oi 6809	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡
107		f1oeq1 6759	. . . . . . . . . . 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 9008	. . . . . . . 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 6759	. . . . . . . . 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 6770	. . . . . . 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 6725	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
118		frn 6662	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
119		xpss1 5643	. . . . . . 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 6731	. . . . . 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 6737	. . . . 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 7807	. . . . . . . 8 ⊢ ∅ ∈ ω
127	126	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ ω)
128	34, 127	sseldd 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ dom 𝑂)
129	73, 128	ffvelcdmd 7022	. . . . 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 9886	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→(ω × 𝑡))
133		f1co 6737	. . . 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 6720	. . . 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 2737	. 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 2737	. 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 2737	. . 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 10491	. 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 2737	. 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 10523	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3062 {crab 3404 Vcvv 3442 [wsbc 3730 ∩ cin 3900 ⊆ wss 3901 ∅c0 4273 ifcif 4477 𝒫 cpw 4551 {csn 4577 ⟨cop 4583 ∪ cuni 4856 ∩ cint 4898 ∪ ciun 4945 class class class wbr 5096 {copab 5158 ↦ cmpt 5179 I cid 5521 E cep 5527 We wwe 5578 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 ∘ ccom 5628 Oncon0 6306 suc csuc 6308 ⟶wf 6479 –1-1→wf1 6480 –1-1-onto→wf1o 6482 ‘cfv 6483 Isom wiso 6484 (class class class)co 7341 ∈ cmpo 7343 ωcom 7784 seq_ωcseqom 8352 ↑_m cmap 8690 ≈ cen 8805 ≼ cdom 8806 ≺ csdm 8807 Fincfn 8808 OrdIsocoi 9370 harchar 9417 cardccrd 9796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-inf2 9502

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4857 df-int 4899 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-se 5580 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-isom 6492 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-seqom 8353 df-1o 8371 df-er 8573 df-map 8692 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-oi 9371 df-har 9418 df-card 9800

This theorem is referenced by: pwfseq 10525

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