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Theorem pwfseqlem5 10078
Description: Lemma for pwfseq 10079. Although in some ways pwfseqlem4 10077 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 9442. 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 9429. 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 9157), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 9433). (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 3447 . . . . . . . . . . 11 𝑡 ∈ V
6 simprl3 1217 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
74, 6sylan2b 596 . . . . . . . . . . 11 ((𝜑𝜓) → 𝑟 We 𝑡)
8 pwfseqlem5.o . . . . . . . . . . . 12 𝑂 = OrdIso(𝑟, 𝑡)
98oiiso 8989 . . . . . . . . . . 11 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
105, 7, 9sylancr 590 . . . . . . . . . 10 ((𝜑𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11 isof1o 7059 . . . . . . . . . 10 (𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡) → 𝑂:dom 𝑂1-1-onto𝑡)
1210, 11syl 17 . . . . . . . . 9 ((𝜑𝜓) → 𝑂:dom 𝑂1-1-onto𝑡)
13 cardom 9403 . . . . . . . . . . . 12 (card‘ω) = ω
14 simprr 772 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
154, 14sylan2b 596 . . . . . . . . . . . . . 14 ((𝜑𝜓) → ω ≼ 𝑡)
168oien 8990 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂𝑡)
175, 7, 16sylancr 590 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → dom 𝑂𝑡)
1817ensymd 8547 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡 ≈ dom 𝑂)
19 domentr 8555 . . . . . . . . . . . . . 14 ((ω ≼ 𝑡𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
2015, 18, 19syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝜓) → ω ≼ dom 𝑂)
21 omelon 9097 . . . . . . . . . . . . . . 15 ω ∈ On
22 onenon 9366 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
2321, 22ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
248oion 8988 . . . . . . . . . . . . . . . 16 (𝑡 ∈ V → dom 𝑂 ∈ On)
2524elv 3449 . . . . . . . . . . . . . . 15 dom 𝑂 ∈ On
26 onenon 9366 . . . . . . . . . . . . . . 15 (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
2725, 26mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝜓) → dom 𝑂 ∈ dom card)
28 carddom2 9394 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
2923, 27, 28sylancr 590 . . . . . . . . . . . . 13 ((𝜑𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
3020, 29mpbird 260 . . . . . . . . . . . 12 ((𝜑𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
3113, 30eqsstrrid 3967 . . . . . . . . . . 11 ((𝜑𝜓) → ω ⊆ (card‘dom 𝑂))
32 cardonle 9374 . . . . . . . . . . . 12 (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
3325, 32mp1i 13 . . . . . . . . . . 11 ((𝜑𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
3431, 33sstrd 3928 . . . . . . . . . 10 ((𝜑𝜓) → ω ⊆ dom 𝑂)
35 sseq2 3944 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36 fveq2 6649 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑁𝑏) = (𝑁‘dom 𝑂))
37 f1oeq1 6583 . . . . . . . . . . . . . 14 ((𝑁𝑏) = (𝑁‘dom 𝑂) → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏))
3836, 37syl 17 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏))
39 xpeq12 5548 . . . . . . . . . . . . . . 15 ((𝑏 = dom 𝑂𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
4039anidms 570 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
4140f1oeq2d 6590 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto𝑏))
42 f1oeq3 6585 . . . . . . . . . . . . 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))
4338, 41, 423bitrd 308 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
4435, 43imbi12d 348 . . . . . . . . . . 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)))
45 pwfseqlem5.n . . . . . . . . . . . 12 (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4645adantr 484 . . . . . . . . . . 11 ((𝜑𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4725a1i 11 . . . . . . . . . . . 12 ((𝜑𝜓) → dom 𝑂 ∈ On)
481adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
49 omex 9094 . . . . . . . . . . . . . . . . . 18 ω ∈ V
50 ovex 7172 . . . . . . . . . . . . . . . . . 18 (𝐴m 𝑛) ∈ V
5149, 50iunex 7655 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
52 f1dmex 7644 . . . . . . . . . . . . . . . . 17 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑛 ∈ ω (𝐴m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
5348, 51, 52sylancl 589 . . . . . . . . . . . . . . . 16 ((𝜑𝜓) → 𝒫 𝐴 ∈ V)
54 pwexb 7472 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
5553, 54sylibr 237 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → 𝐴 ∈ V)
56 simprl1 1215 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡𝐴)
574, 56sylan2b 596 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → 𝑡𝐴)
58 ssdomg 8542 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → (𝑡𝐴𝑡𝐴))
5955, 57, 58sylc 65 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡𝐴)
60 canth2g 8659 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
61 sdomdom 8524 . . . . . . . . . . . . . . 15 (𝐴 ≺ 𝒫 𝐴𝐴 ≼ 𝒫 𝐴)
6255, 60, 613syl 18 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝐴 ≼ 𝒫 𝐴)
63 domtr 8549 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
6459, 62, 63syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑡 ≼ 𝒫 𝐴)
65 endomtr 8554 . . . . . . . . . . . . 13 ((dom 𝑂𝑡𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
6617, 64, 65syl2anc 587 . . . . . . . . . . . 12 ((𝜑𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
67 elharval 9013 . . . . . . . . . . . 12 (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
6847, 66, 67sylanbrc 586 . . . . . . . . . . 11 ((𝜑𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
6944, 46, 68rspcdva 3576 . . . . . . . . . 10 ((𝜑𝜓) → (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
7034, 69mpd 15 . . . . . . . . 9 ((𝜑𝜓) → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)
71 f1oco 6616 . . . . . . . . 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)
7212, 70, 71syl2anc 587 . . . . . . . 8 ((𝜑𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto𝑡)
73 f1of 6594 . . . . . . . . . . . . . . 15 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂𝑡)
7412, 73syl 17 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑂:dom 𝑂𝑡)
7574feqmptd 6712 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)))
76 f1oeq1 6583 . . . . . . . . . . . . 13 (𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡))
7775, 76syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡))
7812, 77mpbid 235 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡)
7974feqmptd 6712 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)))
80 f1oeq1 6583 . . . . . . . . . . . . 13 (𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡))
8179, 80syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡))
8212, 81mpbid 235 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡)
8378, 82xpf1o 8667 . . . . . . . . . 10 ((𝜑𝜓) → (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
84 pwfseqlem5.t . . . . . . . . . . 11 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩)
85 f1oeq1 6583 . . . . . . . . . . 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)))
8684, 85ax-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))
8783, 86sylibr 237 . . . . . . . . 9 ((𝜑𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
88 f1ocnv 6606 . . . . . . . . 9 (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) → 𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
8987, 88syl 17 . . . . . . . 8 ((𝜑𝜓) → 𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
90 f1oco 6616 . . . . . . . 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)
9172, 89, 90syl2anc 587 . . . . . . 7 ((𝜑𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
92 pwfseqlem5.p . . . . . . . 8 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)
93 f1oeq1 6583 . . . . . . . 8 (𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇) → (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡))
9492, 93ax-mp 5 . . . . . . 7 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
9591, 94sylibr 237 . . . . . 6 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1-onto𝑡)
96 f1of1 6593 . . . . . 6 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡𝑃:(𝑡 × 𝑡)–1-1𝑡)
9795, 96syl 17 . . . . 5 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1𝑡)
98 f1of1 6593 . . . . . . . . . . . . 13 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂1-1𝑡)
9912, 98syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝑂:dom 𝑂1-1𝑡)
100 f1ssres 6561 . . . . . . . . . . . 12 ((𝑂:dom 𝑂1-1𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1𝑡)
10199, 34, 100syl2anc 587 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1𝑡)
102 f1f1orn 6605 . . . . . . . . . . 11 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
103101, 102syl 17 . . . . . . . . . 10 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
10474, 34feqresmpt 6713 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂𝑥)))
105 f1oeq1 6583 . . . . . . . . . . 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)))
106104, 105syl 17 . . . . . . . . . 10 ((𝜑𝜓) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
107103, 106mpbid 235 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω))
108 mptresid 5889 . . . . . . . . . . 11 ( I ↾ 𝑡) = (𝑦𝑡𝑦)
109108eqcomi 2810 . . . . . . . . . 10 (𝑦𝑡𝑦) = ( I ↾ 𝑡)
110 f1oi 6631 . . . . . . . . . . 11 ( I ↾ 𝑡):𝑡1-1-onto𝑡
111 f1oeq1 6583 . . . . . . . . . . 11 ((𝑦𝑡𝑦) = ( I ↾ 𝑡) → ((𝑦𝑡𝑦):𝑡1-1-onto𝑡 ↔ ( I ↾ 𝑡):𝑡1-1-onto𝑡))
112110, 111mpbiri 261 . . . . . . . . . 10 ((𝑦𝑡𝑦) = ( I ↾ 𝑡) → (𝑦𝑡𝑦):𝑡1-1-onto𝑡)
113109, 112mp1i 13 . . . . . . . . 9 ((𝜑𝜓) → (𝑦𝑡𝑦):𝑡1-1-onto𝑡)
114107, 113xpf1o 8667 . . . . . . . 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 6583 . . . . . . . . 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 237 . . . . . . 7 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
119 f1of1 6593 . . . . . . 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 6553 . . . . . . 7 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
122 frn 6497 . . . . . . 7 ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
123 xpss1 5542 . . . . . . 7 (ran (𝑂 ↾ ω) ⊆ 𝑡 → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
124101, 121, 122, 1234syl 19 . . . . . 6 ((𝜑𝜓) → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
125 f1ss 6559 . . . . . 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 587 . . . . 5 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
127 f1co 6564 . . . . 5 ((𝑃:(𝑡 × 𝑡)–1-1𝑡𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡)) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
12897, 126, 127syl2anc 587 . . . 4 ((𝜑𝜓) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
1295a1i 11 . . . . 5 ((𝜑𝜓) → 𝑡 ∈ V)
130 peano1 7585 . . . . . . . 8 ∅ ∈ ω
131130a1i 11 . . . . . . 7 ((𝜑𝜓) → ∅ ∈ ω)
13234, 131sseldd 3919 . . . . . 6 ((𝜑𝜓) → ∅ ∈ dom 𝑂)
13374, 132ffvelrnd 6833 . . . . 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, 95, 134, 135fseqenlem2 9440 . . . 4 ((𝜑𝜓) → 𝑄: 𝑛 ∈ ω (𝑡m 𝑛)–1-1→(ω × 𝑡))
137 f1co 6564 . . . 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 587 . . 3 ((𝜑𝜓) → ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡)
139 pwfseqlem5.k . . . 4 𝐾 = ((𝑃𝐼) ∘ 𝑄)
140 f1eq1 6548 . . . 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 237 . 2 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡)
143 eqid 2801 . 2 (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))}) = (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})
144 eqid 2801 . 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 2801 . . 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 10045 . 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 2801 . 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 10077 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  {crab 3113  Vcvv 3444  [wsbc 3723  cin 3883  wss 3884  c0 4246  ifcif 4428  𝒫 cpw 4500  {csn 4528  cop 4534   cuni 4803   cint 4841   ciun 4884   class class class wbr 5033  {copab 5095  cmpt 5113   I cid 5427   E cep 5432   We wwe 5481   × cxp 5521  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  ccom 5527  Oncon0 6163  suc csuc 6165  wf 6324  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328   Isom wiso 6329  (class class class)co 7139  cmpo 7141  ωcom 7564  seqωcseqom 8070  m cmap 8393  cen 8493  cdom 8494  csdm 8495  Fincfn 8496  OrdIsocoi 8961  harchar 9008  cardccrd 9352
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-har 9009  df-card 9356
This theorem is referenced by:  pwfseq  10079
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