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Theorem pwfseqlem5 9523
Description: Lemma for pwfseq 9524. Although in some ways pwfseqlem4 9522 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 8888. 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 8875. 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 8641), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 8879). (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 3234 . . . . . . . . . . 11 𝑡 ∈ V
6 simprl3 1128 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
74, 6sylan2b 491 . . . . . . . . . . 11 ((𝜑𝜓) → 𝑟 We 𝑡)
8 pwfseqlem5.o . . . . . . . . . . . 12 𝑂 = OrdIso(𝑟, 𝑡)
98oiiso 8483 . . . . . . . . . . 11 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
105, 7, 9sylancr 696 . . . . . . . . . 10 ((𝜑𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11 isof1o 6613 . . . . . . . . . 10 (𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡) → 𝑂:dom 𝑂1-1-onto𝑡)
1210, 11syl 17 . . . . . . . . 9 ((𝜑𝜓) → 𝑂:dom 𝑂1-1-onto𝑡)
138oion 8482 . . . . . . . . . . . . 13 (𝑡 ∈ V → dom 𝑂 ∈ On)
145, 13ax-mp 5 . . . . . . . . . . . 12 dom 𝑂 ∈ On
1514a1i 11 . . . . . . . . . . 11 ((𝜑𝜓) → dom 𝑂 ∈ On)
168oien 8484 . . . . . . . . . . . . 13 ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂𝑡)
175, 7, 16sylancr 696 . . . . . . . . . . . 12 ((𝜑𝜓) → dom 𝑂𝑡)
181adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
19 omex 8578 . . . . . . . . . . . . . . . . 17 ω ∈ V
20 ovex 6718 . . . . . . . . . . . . . . . . 17 (𝐴𝑚 𝑛) ∈ V
2119, 20iunex 7189 . . . . . . . . . . . . . . . 16 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
22 f1dmex 7178 . . . . . . . . . . . . . . . 16 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
2318, 21, 22sylancl 695 . . . . . . . . . . . . . . 15 ((𝜑𝜓) → 𝒫 𝐴 ∈ V)
24 pwexb 7017 . . . . . . . . . . . . . . 15 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2523, 24sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝐴 ∈ V)
26 simprl1 1126 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡𝐴)
274, 26sylan2b 491 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡𝐴)
28 ssdomg 8043 . . . . . . . . . . . . . 14 (𝐴 ∈ V → (𝑡𝐴𝑡𝐴))
2925, 27, 28sylc 65 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑡𝐴)
30 canth2g 8155 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
31 sdomdom 8025 . . . . . . . . . . . . . 14 (𝐴 ≺ 𝒫 𝐴𝐴 ≼ 𝒫 𝐴)
3225, 30, 313syl 18 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝐴 ≼ 𝒫 𝐴)
33 domtr 8050 . . . . . . . . . . . . 13 ((𝑡𝐴𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
3429, 32, 33syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝑡 ≼ 𝒫 𝐴)
35 endomtr 8055 . . . . . . . . . . . 12 ((dom 𝑂𝑡𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
3617, 34, 35syl2anc 694 . . . . . . . . . . 11 ((𝜑𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
37 elharval 8509 . . . . . . . . . . 11 (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
3815, 36, 37sylanbrc 699 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
41 cardom 8850 . . . . . . . . . . . 12 (card‘ω) = ω
42 simprr 811 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
434, 42sylan2b 491 . . . . . . . . . . . . . 14 ((𝜑𝜓) → ω ≼ 𝑡)
4417ensymd 8048 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑡 ≈ dom 𝑂)
45 domentr 8056 . . . . . . . . . . . . . 14 ((ω ≼ 𝑡𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
4643, 44, 45syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝜓) → ω ≼ dom 𝑂)
47 omelon 8581 . . . . . . . . . . . . . . 15 ω ∈ On
48 onenon 8813 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
4947, 48ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
50 onenon 8813 . . . . . . . . . . . . . . 15 (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
5114, 50mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝜓) → dom 𝑂 ∈ dom card)
52 carddom2 8841 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
5349, 51, 52sylancr 696 . . . . . . . . . . . . 13 ((𝜑𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
5446, 53mpbird 247 . . . . . . . . . . . 12 ((𝜑𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
5541, 54syl5eqssr 3683 . . . . . . . . . . 11 ((𝜑𝜓) → ω ⊆ (card‘dom 𝑂))
56 cardonle 8821 . . . . . . . . . . . 12 (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
5715, 56syl 17 . . . . . . . . . . 11 ((𝜑𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
5855, 57sstrd 3646 . . . . . . . . . 10 ((𝜑𝜓) → ω ⊆ dom 𝑂)
59 sseq2 3660 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
60 fveq2 6229 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑁𝑏) = (𝑁‘dom 𝑂))
61 f1oeq1 6165 . . . . . . . . . . . . . 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 5168 . . . . . . . . . . . . . . 15 ((𝑏 = dom 𝑂𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
6463anidms 678 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
65 f1oeq2 6166 . . . . . . . . . . . . . 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 6167 . . . . . . . . . . . . 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 333 . . . . . . . . . . 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 3336 . . . . . . . . . 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 6197 . . . . . . . . 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 694 . . . . . . . 8 ((𝜑𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto𝑡)
74 f1of 6175 . . . . . . . . . . . . . . 15 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂𝑡)
7512, 74syl 17 . . . . . . . . . . . . . 14 ((𝜑𝜓) → 𝑂:dom 𝑂𝑡)
7675feqmptd 6288 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂𝑢)))
77 f1oeq1 6165 . . . . . . . . . . . . 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 6288 . . . . . . . . . . . . 13 ((𝜑𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂𝑣)))
81 f1oeq1 6165 . . . . . . . . . . . . 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 8163 . . . . . . . . . 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 6165 . . . . . . . . . . 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 6187 . . . . . . . . 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 6197 . . . . . . . 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 694 . . . . . . 7 ((𝜑𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇):(𝑡 × 𝑡)–1-1-onto𝑡)
93 pwfseqlem5.p . . . . . . . 8 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)
94 f1oeq1 6165 . . . . . . . 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 6174 . . . . . 6 (𝑃:(𝑡 × 𝑡)–1-1-onto𝑡𝑃:(𝑡 × 𝑡)–1-1𝑡)
9896, 97syl 17 . . . . 5 ((𝜑𝜓) → 𝑃:(𝑡 × 𝑡)–1-1𝑡)
99 f1of1 6174 . . . . . . . . . . . . 13 (𝑂:dom 𝑂1-1-onto𝑡𝑂:dom 𝑂1-1𝑡)
10012, 99syl 17 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝑂:dom 𝑂1-1𝑡)
101 f1ssres 6146 . . . . . . . . . . . 12 ((𝑂:dom 𝑂1-1𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1𝑡)
102100, 58, 101syl2anc 694 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω):ω–1-1𝑡)
103 f1f1orn 6186 . . . . . . . . . . 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 6289 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂𝑥)))
106 f1oeq1 6165 . . . . . . . . . . 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 5491 . . . . . . . . . 10 (𝑦𝑡𝑦) = ( I ↾ 𝑡)
110 f1oi 6212 . . . . . . . . . . 11 ( I ↾ 𝑡):𝑡1-1-onto𝑡
111 f1oeq1 6165 . . . . . . . . . . 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 8163 . . . . . . . 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 6165 . . . . . . . . 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 6174 . . . . . . 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 6139 . . . . . . 7 ((𝑂 ↾ ω):ω–1-1𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
122 frn 6091 . . . . . . 7 ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
123 xpss1 5161 . . . . . . 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 6144 . . . . . 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 694 . . . . 5 ((𝜑𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
127 f1co 6148 . . . . 5 ((𝑃:(𝑡 × 𝑡)–1-1𝑡𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡)) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
12898, 126, 127syl2anc 694 . . . 4 ((𝜑𝜓) → (𝑃𝐼):(ω × 𝑡)–1-1𝑡)
1295a1i 11 . . . . 5 ((𝜑𝜓) → 𝑡 ∈ V)
130 peano1 7127 . . . . . . . 8 ∅ ∈ ω
131130a1i 11 . . . . . . 7 ((𝜑𝜓) → ∅ ∈ ω)
13258, 131sseldd 3637 . . . . . 6 ((𝜑𝜓) → ∅ ∈ dom 𝑂)
13375, 132ffvelrnd 6400 . . . . 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 8886 . . . 4 ((𝜑𝜓) → 𝑄: 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1→(ω × 𝑡))
137 f1co 6148 . . . 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 694 . . 3 ((𝜑𝜓) → ((𝑃𝐼) ∘ 𝑄): 𝑛 ∈ ω (𝑡𝑚 𝑛)–1-1𝑡)
139 pwfseqlem5.k . . . 4 𝐾 = ((𝑃𝐼) ∘ 𝑄)
140 f1eq1 6134 . . . 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 2651 . 2 (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))}) = (𝐺‘{𝑖𝑡 ∣ ((𝐾𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (𝐺‘(𝐾𝑖)))})
144 eqid 2651 . 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 2651 . . 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 9490 . 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 2651 . 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 9522 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  [wsbc 3468  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   cint 4507   ciun 4552   class class class wbr 4685  {copab 4745  cmpt 4762   I cid 5052   E cep 5057   We wwe 5101   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  ccom 5147  Oncon0 5761  suc csuc 5763  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  cmpt2 6692  ωcom 7107  seq𝜔cseqom 7587  𝑚 cmap 7899  cen 7994  cdom 7995  csdm 7996  Fincfn 7997  OrdIsocoi 8455  harchar 8502  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803
This theorem is referenced by:  pwfseq  9524
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