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

Theorem pwfseqlem5 9470
Description: Lemma for pwfseq 9471. Although in some ways pwfseqlem4 9469 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 8835. 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 8822. 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 8588), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 8826). (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses
Ref Expression
pwfseqlem5.g (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
pwfseqlem5.x (𝜑𝑋𝐴)
pwfseqlem5.h (𝜑𝐻:ω–1-1-onto𝑋)
pwfseqlem5.ps (𝜓 ↔ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
pwfseqlem5.n (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
pwfseqlem5.o 𝑂 = OrdIso(𝑟, 𝑡)
pwfseqlem5.t 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩)
pwfseqlem5.p 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)
pwfseqlem5.s 𝑆 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡𝑚 suc 𝑘) ↦ ((𝑓‘(𝑥𝑘))𝑃(𝑥𝑘)))), {⟨∅, (𝑂‘∅)⟩})
pwfseqlem5.q 𝑄 = (𝑦 𝑛 ∈ ω (𝑡𝑚 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
pwfseqlem5.i 𝐼 = (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩)
pwfseqlem5.k 𝐾 = ((𝑃𝐼) ∘ 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.
StepHypRef Expression
1 pwfseqlem5.g . 2 (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
2 pwfseqlem5.x . 2 (𝜑𝑋𝐴)
3 pwfseqlem5.h . 2 (𝜑𝐻:ω–1-1-onto𝑋)
4 pwfseqlem5.ps . 2 (𝜓 ↔ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
5 vex 3198 . . . . . . . . . . 11 𝑡 ∈ V
6 simprl3 1106 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
74, 6sylan2b 492 . . . . . . . . . . 11 ((𝜑𝜓) → 𝑟 We 𝑡)
8 pwfseqlem5.o . . . . . . . . . . . 12 𝑂 = OrdIso(𝑟, 𝑡)
98oiiso 8427 . . . . . . . . . . 11 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
105, 7, 9sylancr 694 . . . . . . . . . 10 ((𝜑𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11 isof1o 6558 . . . . . . . . . 10 (𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡) → 𝑂:dom 𝑂1-1-onto𝑡)
1210, 11syl 17 . . . . . . . . 9 ((𝜑𝜓) → 𝑂:dom 𝑂1-1-onto𝑡)
138oion 8426 . . . . . . . . . . . . 13 (𝑡 ∈ V → dom 𝑂 ∈ On)
145, 13ax-mp 5 . . . . . . . . . . . 12 dom 𝑂 ∈ On
1514a1i 11 . . . . . . . . . . 11 ((𝜑𝜓) → dom 𝑂 ∈ On)
168oien 8428 . . . . . . . . . . . . 13 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂𝑡)
175, 7, 16sylancr 694 . . . . . . . . . . . 12 ((𝜑𝜓) → dom 𝑂𝑡)
181adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
19 omex 8525 . . . . . . . . . . . . . . . . 17 ω ∈ V
20 ovex 6663 . . . . . . . . . . . . . . . . 17 (𝐴𝑚 𝑛) ∈ V
2119, 20iunex 7132 . . . . . . . . . . . . . . . 16 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
22 f1dmex 7121 . . . . . . . . . . . . . . . 16 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
2318, 21, 22sylancl 693 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → 𝒫 𝐴 ∈ V)
24 pwexb 6960 . . . . . . . . . . . . . . 15 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2523, 24sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝐴 ∈ V)
26 simprl1 1104 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡𝐴)
274, 26sylan2b 492 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡𝐴)
28 ssdomg 7986 . . . . . . . . . . . . . 14 (𝐴 ∈ V → (𝑡𝐴𝑡𝐴))
2925, 27, 28sylc 65 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑡𝐴)
30 canth2g 8099 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
31 sdomdom 7968 . . . . . . . . . . . . . 14 (𝐴 ≺ 𝒫 𝐴𝐴 ≼ 𝒫 𝐴)
3225, 30, 313syl 18 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝐴 ≼ 𝒫 𝐴)
33 domtr 7994 . . . . . . . . . . . . 13 ((𝑡𝐴𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
3429, 32, 33syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝑡 ≼ 𝒫 𝐴)
35 endomtr 7999 . . . . . . . . . . . 12 ((dom 𝑂𝑡𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
3617, 34, 35syl2anc 692 . . . . . . . . . . 11 ((𝜑𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
37 elharval 8453 . . . . . . . . . . 11 (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
3815, 36, 37sylanbrc 697 . . . . . . . . . 10 ((𝜑𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
39 pwfseqlem5.n . . . . . . . . . . 11 (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4039adantr 481 . . . . . . . . . 10 ((𝜑𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
41 cardom 8797 . . . . . . . . . . . 12 (card‘ω) = ω
42 simprr 795 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
434, 42sylan2b 492 . . . . . . . . . . . . . 14 ((𝜑𝜓) → ω ≼ 𝑡)
4417ensymd 7992 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡 ≈ dom 𝑂)
45 domentr 8000 . . . . . . . . . . . . . 14 ((ω ≼ 𝑡𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
4643, 44, 45syl2anc 692 . . . . . . . . . . . . 13 ((𝜑𝜓) → ω ≼ dom 𝑂)
47 omelon 8528 . . . . . . . . . . . . . . 15 ω ∈ On
48 onenon 8760 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
4947, 48ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
50 onenon 8760 . . . . . . . . . . . . . . 15 (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
5114, 50mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝜓) → dom 𝑂 ∈ dom card)
52 carddom2 8788 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
5349, 51, 52sylancr 694 . . . . . . . . . . . . 13 ((𝜑𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
5446, 53mpbird 247 . . . . . . . . . . . 12 ((𝜑𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
5541, 54syl5eqssr 3642 . . . . . . . . . . 11 ((𝜑𝜓) → ω ⊆ (card‘dom 𝑂))
56 cardonle 8768 . . . . . . . . . . . 12 (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
5715, 56syl 17 . . . . . . . . . . 11 ((𝜑𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
5855, 57sstrd 3605 . . . . . . . . . 10 ((𝜑𝜓) → ω ⊆ dom 𝑂)
59 sseq2 3619 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
60 fveq2 6178 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑁𝑏) = (𝑁‘dom 𝑂))
61 f1oeq1 6114 . . . . . . . . . . . . . 14 ((𝑁𝑏) = (𝑁‘dom 𝑂) → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏))
6260, 61syl 17 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏))
63 xpeq12 5124 . . . . . . . . . . . . . . 15 ((𝑏 = dom 𝑂𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
6463anidms 676 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
65 f1oeq2 6115 . . . . . . . . . . . . . 14 ((𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂) → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto𝑏))
6664, 65syl 17 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto𝑏))
67 f1oeq3 6116 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
6862, 66, 673bitrd 294 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
6959, 68imbi12d 334 . . . . . . . . . . 11 (𝑏 = dom 𝑂 → ((ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)))
7069rspcv 3300 . . . . . . . . . 10 (dom 𝑂 ∈ (har‘𝒫 𝐴) → (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) → (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)))
7138, 40, 58, 70syl3c 66 . . . . . . . . 9 ((𝜑𝜓) → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)
72 f1oco 6146 . . . . . . . . 9 ((𝑂:dom 𝑂1-1-onto𝑡 ∧ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto𝑡)
7312, 71, 72syl2anc 692 . . . . . . . 8 ((𝜑𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto𝑡)
74 f1of 6124 . . . . . . . . . . . . . . 15 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂𝑡)
7512, 74syl 17 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑂:dom 𝑂𝑡)
7675feqmptd 6236 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)))
77 f1oeq1 6114 . . . . . . . . . . . . 13 (𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡))
7876, 77syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡))
7912, 78mpbid 222 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡)
8075feqmptd 6236 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)))
81 f1oeq1 6114 . . . . . . . . . . . . 13 (𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡))
8280, 81syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡))
8312, 82mpbid 222 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡)
8479, 83xpf1o 8107 . . . . . . . . . 10 ((𝜑𝜓) → (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
85 pwfseqlem5.t . . . . . . . . . . 11 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩)
86 f1oeq1 6114 . . . . . . . . . . 11 (𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩) → (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡)))
8785, 86ax-mp 5 . . . . . . . . . 10 (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
8884, 87sylibr 224 . . . . . . . . 9 ((𝜑𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
89 f1ocnv 6136 . . . . . . . . 9 (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) → 𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
9088, 89syl 17 . . . . . . . 8 ((𝜑𝜓) → 𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
91 f1oco 6146 . . . . . . . 8 (((𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto𝑡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂)) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
9273, 90, 91syl2anc 692 . . . . . . 7 ((𝜑𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
93 pwfseqlem5.p . . . . . . . 8 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)
94 f1oeq1 6114 . . . . . . . 8 (𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇) → (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡))
9593, 94ax-mp 5 . . . . . . 7 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
9692, 95sylibr 224 . . . . . 6 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1-onto𝑡)
97 f1of1 6123 . . . . . 6 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡𝑃:(𝑡 × 𝑡)–1-1𝑡)
9896, 97syl 17 . . . . 5 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1𝑡)
99 f1of1 6123 . . . . . . . . . . . . 13 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂1-1𝑡)
10012, 99syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝑂:dom 𝑂1-1𝑡)
101 f1ssres 6095 . . . . . . . . . . . 12 ((𝑂:dom 𝑂1-1𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1𝑡)
102100, 58, 101syl2anc 692 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1𝑡)
103 f1f1orn 6135 . . . . . . . . . . 11 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
104102, 103syl 17 . . . . . . . . . 10 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
10575, 58feqresmpt 6237 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂𝑥)))
106 f1oeq1 6114 . . . . . . . . . . 11 ((𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂𝑥)) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
107105, 106syl 17 . . . . . . . . . 10 ((𝜑𝜓) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
108104, 107mpbid 222 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω))
109 mptresid 5444 . . . . . . . . . 10 (𝑦𝑡𝑦) = ( I ↾ 𝑡)
110 f1oi 6161 . . . . . . . . . . 11 ( I ↾ 𝑡):𝑡1-1-onto𝑡
111 f1oeq1 6114 . . . . . . . . . . 11 ((𝑦𝑡𝑦) = ( I ↾ 𝑡) → ((𝑦𝑡𝑦):𝑡1-1-onto𝑡 ↔ ( I ↾ 𝑡):𝑡1-1-onto𝑡))
112110, 111mpbiri 248 . . . . . . . . . 10 ((𝑦𝑡𝑦) = ( I ↾ 𝑡) → (𝑦𝑡𝑦):𝑡1-1-onto𝑡)
113109, 112mp1i 13 . . . . . . . . 9 ((𝜑𝜓) → (𝑦𝑡𝑦):𝑡1-1-onto𝑡)
114108, 113xpf1o 8107 . . . . . . . 8 ((𝜑𝜓) → (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
115 pwfseqlem5.i . . . . . . . . 9 𝐼 = (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩)
116 f1oeq1 6114 . . . . . . . . 9 (𝐼 = (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩) → (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡)))
117115, 116ax-mp 5 . . . . . . . 8 (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
118114, 117sylibr 224 . . . . . . 7 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
119 f1of1 6123 . . . . . . 7 (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
120118, 119syl 17 . . . . . 6 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
121 f1f 6088 . . . . . . 7 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
122 frn 6040 . . . . . . 7 ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
123 xpss1 5218 . . . . . . 7 (ran (𝑂 ↾ ω) ⊆ 𝑡 → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
124102, 121, 122, 1234syl 19 . . . . . 6 ((𝜑𝜓) → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
125 f1ss 6093 . . . . . 6 ((𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡) ∧ (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡)) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
126120, 124, 125syl2anc 692 . . . . 5 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
127 f1co 6097 . . . . 5 ((𝑃:(𝑡 × 𝑡)–1-1𝑡𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡)) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
12898, 126, 127syl2anc 692 . . . 4 ((𝜑𝜓) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
1295a1i 11 . . . . 5 ((𝜑𝜓) → 𝑡 ∈ V)
130 peano1 7070 . . . . . . . 8 ∅ ∈ ω
131130a1i 11 . . . . . . 7 ((𝜑𝜓) → ∅ ∈ ω)
13258, 131sseldd 3596 . . . . . 6 ((𝜑𝜓) → ∅ ∈ dom 𝑂)
13375, 132ffvelrnd 6346 . . . . 5 ((𝜑𝜓) → (𝑂‘∅) ∈ 𝑡)
134 pwfseqlem5.s . . . . 5 𝑆 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡𝑚 suc 𝑘) ↦ ((𝑓‘(𝑥𝑘))𝑃(𝑥𝑘)))), {⟨∅, (𝑂‘∅)⟩})
135 pwfseqlem5.q . . . . 5 𝑄 = (𝑦 𝑛 ∈ ω (𝑡𝑚 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
136129, 133, 96, 134, 135fseqenlem2 8833 . . . 4 ((𝜑𝜓) → 𝑄: 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1→(ω × 𝑡))
137 f1co 6097 . . . 4 (((𝑃𝐼):(ω × 𝑡)–1-1𝑡𝑄: 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1→(ω × 𝑡)) → ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡)
138128, 136, 137syl2anc 692 . . 3 ((𝜑𝜓) → ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡)
139 pwfseqlem5.k . . . 4 𝐾 = ((𝑃𝐼) ∘ 𝑄)
140 f1eq1 6083 . . . 4 (𝐾 = ((𝑃𝐼) ∘ 𝑄) → (𝐾: 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡 ↔ ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡))
141139, 140ax-mp 5 . . 3 (𝐾: 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡 ↔ ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡)
142138, 141sylibr 224 . 2 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡)
143 eqid 2620 . 2 (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))}) = (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})
144 eqid 2620 . 2 (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡}))) = (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))
145 eqid 2620 . . 3 {⟨𝑐, 𝑑⟩ ∣ ((𝑐𝐴𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚𝑐 [(𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐𝐴𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚𝑐 [(𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
146145fpwwe2cbv 9437 . 2 {⟨𝑐, 𝑑⟩ ∣ ((𝑐𝐴𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚𝑐 [(𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏𝑎 [(𝑠 “ {𝑏}) / 𝑤](𝑤(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))(𝑠 ∩ (𝑤 × 𝑤))) = 𝑏))}
147 eqid 2620 . 2 dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐𝐴𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚𝑐 [(𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐𝐴𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚𝑐 [(𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
1481, 2, 3, 4, 142, 143, 144, 146, 147pwfseqlem4 9469 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  [wsbc 3429  cin 3566  wss 3567  c0 3907  ifcif 4077  𝒫 cpw 4149  {csn 4168  cop 4174   cuni 4427   cint 4466   ciun 4511   class class class wbr 4644  {copab 4703  cmpt 4720   I cid 5013   E cep 5018   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  ran crn 5105  cres 5106  cima 5107  ccom 5108  Oncon0 5711  suc csuc 5713  wf 5872  1-1wf1 5873  1-1-ontowf1o 5875  cfv 5876   Isom wiso 5877  (class class class)co 6635  cmpt2 6637  ωcom 7050  seq𝜔cseqom 7527  𝑚 cmap 7842  cen 7937  cdom 7938  csdm 7939  Fincfn 7940  OrdIsocoi 8399  harchar 8446  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750
This theorem is referenced by:  pwfseq  9471
  Copyright terms: Public domain W3C validator