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Theorem pwfseqlem5 10703
Description: Lemma for pwfseq 10704. Although in some ways pwfseqlem4 10702 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 10067. 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 10054. 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 9746), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 10058). (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 3484 . . . . . . . . . . 11 𝑡 ∈ V
6 simprl3 1221 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
74, 6sylan2b 594 . . . . . . . . . . 11 ((𝜑𝜓) → 𝑟 We 𝑡)
8 pwfseqlem5.o . . . . . . . . . . . 12 𝑂 = OrdIso(𝑟, 𝑡)
98oiiso 9577 . . . . . . . . . . 11 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
105, 7, 9sylancr 587 . . . . . . . . . 10 ((𝜑𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11 isof1o 7343 . . . . . . . . . 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 10026 . . . . . . . . . . . 12 (card‘ω) = ω
14 simprr 773 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
154, 14sylan2b 594 . . . . . . . . . . . . . 14 ((𝜑𝜓) → ω ≼ 𝑡)
168oien 9578 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂𝑡)
175, 7, 16sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → dom 𝑂𝑡)
1817ensymd 9045 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡 ≈ dom 𝑂)
19 domentr 9053 . . . . . . . . . . . . . 14 ((ω ≼ 𝑡𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
2015, 18, 19syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝜓) → ω ≼ dom 𝑂)
21 omelon 9686 . . . . . . . . . . . . . . 15 ω ∈ On
22 onenon 9989 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
2321, 22ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
248oion 9576 . . . . . . . . . . . . . . . 16 (𝑡 ∈ V → dom 𝑂 ∈ On)
2524elv 3485 . . . . . . . . . . . . . . 15 dom 𝑂 ∈ On
26 onenon 9989 . . . . . . . . . . . . . . 15 (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
2725, 26mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝜓) → dom 𝑂 ∈ dom card)
28 carddom2 10017 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
2923, 27, 28sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
3020, 29mpbird 257 . . . . . . . . . . . 12 ((𝜑𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
3113, 30eqsstrrid 4023 . . . . . . . . . . 11 ((𝜑𝜓) → ω ⊆ (card‘dom 𝑂))
32 cardonle 9997 . . . . . . . . . . . 12 (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
3325, 32mp1i 13 . . . . . . . . . . 11 ((𝜑𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
3431, 33sstrd 3994 . . . . . . . . . 10 ((𝜑𝜓) → ω ⊆ dom 𝑂)
35 sseq2 4010 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
36 fveq2 6906 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑁𝑏) = (𝑁‘dom 𝑂))
3736f1oeq1d 6843 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏))
38 xpeq12 5710 . . . . . . . . . . . . . . 15 ((𝑏 = dom 𝑂𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
3938anidms 566 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
4039f1oeq2d 6844 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto𝑏))
41 f1oeq3 6838 . . . . . . . . . . . . 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))
4237, 40, 413bitrd 305 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → ((𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
4335, 42imbi12d 344 . . . . . . . . . . 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)))
44 pwfseqlem5.n . . . . . . . . . . . 12 (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4544adantr 480 . . . . . . . . . . 11 ((𝜑𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4625a1i 11 . . . . . . . . . . . 12 ((𝜑𝜓) → dom 𝑂 ∈ On)
471adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
48 omex 9683 . . . . . . . . . . . . . . . . . 18 ω ∈ V
49 ovex 7464 . . . . . . . . . . . . . . . . . 18 (𝐴m 𝑛) ∈ V
5048, 49iunex 7993 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
51 f1dmex 7981 . . . . . . . . . . . . . . . . 17 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑛 ∈ ω (𝐴m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
5247, 50, 51sylancl 586 . . . . . . . . . . . . . . . 16 ((𝜑𝜓) → 𝒫 𝐴 ∈ V)
53 pwexb 7786 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
5452, 53sylibr 234 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → 𝐴 ∈ V)
55 simprl1 1219 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡𝐴)
564, 55sylan2b 594 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → 𝑡𝐴)
57 ssdomg 9040 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → (𝑡𝐴𝑡𝐴))
5854, 56, 57sylc 65 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡𝐴)
59 canth2g 9171 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
60 sdomdom 9020 . . . . . . . . . . . . . . 15 (𝐴 ≺ 𝒫 𝐴𝐴 ≼ 𝒫 𝐴)
6154, 59, 603syl 18 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝐴 ≼ 𝒫 𝐴)
62 domtr 9047 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
6358, 61, 62syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑡 ≼ 𝒫 𝐴)
64 endomtr 9052 . . . . . . . . . . . . 13 ((dom 𝑂𝑡𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
6517, 63, 64syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
66 elharval 9601 . . . . . . . . . . . 12 (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
6746, 65, 66sylanbrc 583 . . . . . . . . . . 11 ((𝜑𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
6843, 45, 67rspcdva 3623 . . . . . . . . . 10 ((𝜑𝜓) → (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
6934, 68mpd 15 . . . . . . . . 9 ((𝜑𝜓) → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)
70 f1oco 6871 . . . . . . . . 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)
7112, 69, 70syl2anc 584 . . . . . . . 8 ((𝜑𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto𝑡)
72 f1of 6848 . . . . . . . . . . . . . . 15 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂𝑡)
7312, 72syl 17 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑂:dom 𝑂𝑡)
7473feqmptd 6977 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)))
7574f1oeq1d 6843 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡))
7612, 75mpbid 232 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)):dom 𝑂1-1-onto𝑡)
7773feqmptd 6977 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)))
7877f1oeq1d 6843 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑂:dom 𝑂1-1-onto𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡))
7912, 78mpbid 232 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)):dom 𝑂1-1-onto𝑡)
8076, 79xpf1o 9179 . . . . . . . . . 10 ((𝜑𝜓) → (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
81 pwfseqlem5.t . . . . . . . . . . 11 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩)
82 f1oeq1 6836 . . . . . . . . . . 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)))
8381, 82ax-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))
8480, 83sylibr 234 . . . . . . . . 9 ((𝜑𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
85 f1ocnv 6860 . . . . . . . . 9 (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) → 𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
8684, 85syl 17 . . . . . . . 8 ((𝜑𝜓) → 𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
87 f1oco 6871 . . . . . . . 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)
8871, 86, 87syl2anc 584 . . . . . . 7 ((𝜑𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
89 pwfseqlem5.p . . . . . . . 8 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)
90 f1oeq1 6836 . . . . . . . 8 (𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇) → (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡))
9189, 90ax-mp 5 . . . . . . 7 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
9288, 91sylibr 234 . . . . . 6 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1-onto𝑡)
93 f1of1 6847 . . . . . 6 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡𝑃:(𝑡 × 𝑡)–1-1𝑡)
9492, 93syl 17 . . . . 5 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1𝑡)
95 f1of1 6847 . . . . . . . . . . . . 13 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂1-1𝑡)
9612, 95syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝑂:dom 𝑂1-1𝑡)
97 f1ssres 6811 . . . . . . . . . . . 12 ((𝑂:dom 𝑂1-1𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1𝑡)
9896, 34, 97syl2anc 584 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1𝑡)
99 f1f1orn 6859 . . . . . . . . . . 11 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
10098, 99syl 17 . . . . . . . . . 10 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
10173, 34feqresmpt 6978 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂𝑥)))
102101f1oeq1d 6843 . . . . . . . . . 10 ((𝜑𝜓) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
103100, 102mpbid 232 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 ∈ ω ↦ (𝑂𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω))
104 mptresid 6069 . . . . . . . . . . 11 ( I ↾ 𝑡) = (𝑦𝑡𝑦)
105104eqcomi 2746 . . . . . . . . . 10 (𝑦𝑡𝑦) = ( I ↾ 𝑡)
106 f1oi 6886 . . . . . . . . . . 11 ( I ↾ 𝑡):𝑡1-1-onto𝑡
107 f1oeq1 6836 . . . . . . . . . . 11 ((𝑦𝑡𝑦) = ( I ↾ 𝑡) → ((𝑦𝑡𝑦):𝑡1-1-onto𝑡 ↔ ( I ↾ 𝑡):𝑡1-1-onto𝑡))
108106, 107mpbiri 258 . . . . . . . . . 10 ((𝑦𝑡𝑦) = ( I ↾ 𝑡) → (𝑦𝑡𝑦):𝑡1-1-onto𝑡)
109105, 108mp1i 13 . . . . . . . . 9 ((𝜑𝜓) → (𝑦𝑡𝑦):𝑡1-1-onto𝑡)
110103, 109xpf1o 9179 . . . . . . . 8 ((𝜑𝜓) → (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
111 pwfseqlem5.i . . . . . . . . 9 𝐼 = (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩)
112 f1oeq1 6836 . . . . . . . . 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)))
113111, 112ax-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))
114110, 113sylibr 234 . . . . . . 7 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
115 f1of1 6847 . . . . . . 7 (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
116114, 115syl 17 . . . . . 6 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
117 f1f 6804 . . . . . . 7 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
118 frn 6743 . . . . . . 7 ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
119 xpss1 5704 . . . . . . 7 (ran (𝑂 ↾ ω) ⊆ 𝑡 → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
12098, 117, 118, 1194syl 19 . . . . . 6 ((𝜑𝜓) → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
121 f1ss 6809 . . . . . 6 ((𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡) ∧ (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡)) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
122116, 120, 121syl2anc 584 . . . . 5 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
123 f1co 6815 . . . . 5 ((𝑃:(𝑡 × 𝑡)–1-1𝑡𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡)) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
12494, 122, 123syl2anc 584 . . . 4 ((𝜑𝜓) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
1255a1i 11 . . . . 5 ((𝜑𝜓) → 𝑡 ∈ V)
126 peano1 7910 . . . . . . . 8 ∅ ∈ ω
127126a1i 11 . . . . . . 7 ((𝜑𝜓) → ∅ ∈ ω)
12834, 127sseldd 3984 . . . . . 6 ((𝜑𝜓) → ∅ ∈ dom 𝑂)
12973, 128ffvelcdmd 7105 . . . . 5 ((𝜑𝜓) → (𝑂‘∅) ∈ 𝑡)
130 pwfseqlem5.s . . . . 5 𝑆 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡m suc 𝑘) ↦ ((𝑓‘(𝑥𝑘))𝑃(𝑥𝑘)))), {⟨∅, (𝑂‘∅)⟩})
131 pwfseqlem5.q . . . . 5 𝑄 = (𝑦 𝑛 ∈ ω (𝑡m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
132125, 129, 92, 130, 131fseqenlem2 10065 . . . 4 ((𝜑𝜓) → 𝑄: 𝑛 ∈ ω (𝑡m 𝑛)–1-1→(ω × 𝑡))
133 f1co 6815 . . . 4 (((𝑃𝐼):(ω × 𝑡)–1-1𝑡𝑄: 𝑛 ∈ ω (𝑡m 𝑛)–1-1→(ω × 𝑡)) → ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡)
134124, 132, 133syl2anc 584 . . 3 ((𝜑𝜓) → ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡)
135 pwfseqlem5.k . . . 4 𝐾 = ((𝑃𝐼) ∘ 𝑄)
136 f1eq1 6799 . . . 4 (𝐾 = ((𝑃𝐼) ∘ 𝑄) → (𝐾: 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡 ↔ ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡))
137135, 136ax-mp 5 . . 3 (𝐾: 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡 ↔ ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡)
138134, 137sylibr 234 . 2 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑡m 𝑛)–1-1𝑡)
139 eqid 2737 . 2 (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))}) = (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})
140 eqid 2737 . 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})))
141 eqid 2737 . . 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))}
142141fpwwe2cbv 10670 . 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))}
143 eqid 2737 . 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))}
1441, 2, 3, 4, 138, 139, 140, 142, 143pwfseqlem4 10702 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  [wsbc 3788  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  {csn 4626  cop 4632   cuni 4907   cint 4946   ciun 4991   class class class wbr 5143  {copab 5205  cmpt 5225   I cid 5577   E cep 5583   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Oncon0 6384  suc csuc 6386  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561   Isom wiso 6562  (class class class)co 7431  cmpo 7433  ωcom 7887  seqωcseqom 8487  m cmap 8866  cen 8982  cdom 8983  csdm 8984  Fincfn 8985  OrdIsocoi 9549  harchar 9596  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979
This theorem is referenced by:  pwfseq  10704
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