pwfseqlem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pwfseqlem5		Structured version Visualization version GIF version

Theorem pwfseqlem5 10546

Description: Lemma for pwfseq 10547. Although in some ways pwfseqlem4 10545 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 9910. 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 9897. 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 9591), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 9901). (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 3438	. . . . . . . . . . 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 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑟 We 𝑡)
8		pwfseqlem5.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑟, 𝑡)
9	8	oiiso 9418	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
10	5, 7, 9	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11		isof1o 7252	. . . . . . . . . 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 9871	. . . . . . . . . . . 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 594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑡)
16	8	oien 9419	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂 ≈ 𝑡)
17	5, 7, 16	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≈ 𝑡)
18	17	ensymd 8922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≈ dom 𝑂)
19		domentr 8930	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝑡 ∧ 𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
20	15, 18, 19	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ dom 𝑂)
21		omelon 9531	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
22		onenon 9834	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
24	8	oion 9417	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ V → dom 𝑂 ∈ On)
25	24	elv 3439	. . . . . . . . . . . . . . 15 ⊢ dom 𝑂 ∈ On
26		onenon 9834	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
27	25, 26	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ dom card)
28		carddom2 9862	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
29	23, 27, 28	sylancr 587	. . . . . . . . . . . . 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 3972	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ (card‘dom 𝑂))
32		cardonle 9842	. . . . . . . . . . . 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 3943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ dom 𝑂)
35		sseq2 3959	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36		fveq2 6817	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑁‘𝑏) = (𝑁‘dom 𝑂))
37	36	f1oeq1d 6754	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏))
38		xpeq12 5639	. . . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . 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 6749	. . . . . . . . . . . . 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 9528	. . . . . . . . . . . . . . . . . 18 ⊢ ω ∈ V
49		ovex 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ↑_m 𝑛) ∈ V
50	48, 49	iunex 7895	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
51		f1dmex 7884	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝒫 𝐴 ∈ V)
53		pwexb 7694	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
54	52, 53	sylibr 234	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ⊆ 𝐴)
57		ssdomg 8917	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → (𝑡 ⊆ 𝐴 → 𝑡 ≼ 𝐴))
58	54, 56, 57	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝐴)
59		canth2g 9039	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
60		sdomdom 8897	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
61	54, 59, 60	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≼ 𝒫 𝐴)
62		domtr 8924	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
63	58, 61, 62	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝒫 𝐴)
64		endomtr 8929	. . . . . . . . . . . . 13 ⊢ ((dom 𝑂 ≈ 𝑡 ∧ 𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
65	17, 63, 64	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
66		elharval 9442	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
67	46, 65, 66	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
68	43, 45, 67	rspcdva 3576	. . . . . . . . . 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 6782	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡)
72		f1of 6759	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)))
75	74	f1oeq1d 6754	. . . . . . . . . . . 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 6885	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)))
78	77	f1oeq1d 6754	. . . . . . . . . . . 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 9047	. . . . . . . . . 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 6747	. . . . . . . . . . 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 6771	. . . . . . . . 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 6782	. . . . . . . 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 584	. . . . . . 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 6747	. . . . . . . 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 6758	. . . . . 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 6758	. . . . . . . . . . . . 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 6722	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂–1-1→𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1→𝑡)
98	96, 34, 97	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω–1-1→𝑡)
99		f1f1orn 6770	. . . . . . . . . . 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 6886	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)))
102	101	f1oeq1d 6754	. . . . . . . . . 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 5997	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡) = (𝑦 ∈ 𝑡 ↦ 𝑦)
105	104	eqcomi 2739	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡)
106		f1oi 6797	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡
107		f1oeq1 6747	. . . . . . . . . . 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 9047	. . . . . . . 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 6747	. . . . . . . . 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 6758	. . . . . . 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 6715	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
118		frn 6654	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
119		xpss1 5633	. . . . . . 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 6720	. . . . . 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 584	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
123		f1co 6726	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡)
125	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ∈ V)
126		peano1 7814	. . . . . . . 8 ⊢ ∅ ∈ ω
127	126	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ ω)
128	34, 127	sseldd 3933	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ dom 𝑂)
129	73, 128	ffvelcdmd 7013	. . . . 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 9908	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→(ω × 𝑡))
133		f1co 6726	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑_m 𝑛)–1-1→𝑡)
135		pwfseqlem5.k	. . . 4 ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
136		f1eq1 6710	. . . 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 2730	. 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 2730	. 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 2730	. . 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 10513	. 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 2730	. 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 10545	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ∀wral 3045 {crab 3393 Vcvv 3434 [wsbc 3739 ∩ cin 3899 ⊆ wss 3900 ∅c0 4281 ifcif 4473 𝒫 cpw 4548 {csn 4574 ⟨cop 4580 ∪ cuni 4857 ∩ cint 4895 ∪ ciun 4939 class class class wbr 5089 {copab 5151 ↦ cmpt 5170 I cid 5508 E cep 5513 We wwe 5566 × cxp 5612 ^◡ccnv 5613 dom cdm 5614 ran crn 5615 ↾ cres 5616 “ cima 5617 ∘ ccom 5618 Oncon0 6302 suc csuc 6304 ⟶wf 6473 –1-1→wf1 6474 –1-1-onto→wf1o 6476 ‘cfv 6477 Isom wiso 6478 (class class class)co 7341 ∈ cmpo 7343 ωcom 7791 seq_ωcseqom 8361 ↑_m cmap 8745 ≈ cen 8861 ≼ cdom 8862 ≺ csdm 8863 Fincfn 8864 OrdIsocoi 9390 harchar 9437 cardccrd 9820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-seqom 8362 df-1o 8380 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-oi 9391 df-har 9438 df-card 9824

This theorem is referenced by: pwfseq 10547

W3C validator