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Theorem pwfseqlem5 10604
Description: Lemma for pwfseq 10605. Although in some ways pwfseqlem4 10603 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 π‘₯ to π‘₯. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 9968. 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 9955. 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 9647), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 9959). (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses
Ref Expression
pwfseqlem5.g (πœ‘ β†’ 𝐺:𝒫 𝐴–1-1β†’βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
pwfseqlem5.x (πœ‘ β†’ 𝑋 βŠ† 𝐴)
pwfseqlem5.h (πœ‘ β†’ 𝐻:ω–1-1-onto→𝑋)
pwfseqlem5.ps (πœ“ ↔ ((𝑑 βŠ† 𝐴 ∧ π‘Ÿ βŠ† (𝑑 Γ— 𝑑) ∧ π‘Ÿ We 𝑑) ∧ Ο‰ β‰Ό 𝑑))
pwfseqlem5.n (πœ‘ β†’ βˆ€π‘ ∈ (harβ€˜π’« 𝐴)(Ο‰ βŠ† 𝑏 β†’ (π‘β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
pwfseqlem5.o 𝑂 = OrdIso(π‘Ÿ, 𝑑)
pwfseqlem5.t 𝑇 = (𝑒 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(π‘‚β€˜π‘’), (π‘‚β€˜π‘£)⟩)
pwfseqlem5.p 𝑃 = ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇)
pwfseqlem5.s 𝑆 = seqΟ‰((π‘˜ ∈ V, 𝑓 ∈ V ↦ (π‘₯ ∈ (𝑑 ↑m suc π‘˜) ↦ ((π‘“β€˜(π‘₯ β†Ύ π‘˜))𝑃(π‘₯β€˜π‘˜)))), {βŸ¨βˆ…, (π‘‚β€˜βˆ…)⟩})
pwfseqlem5.q 𝑄 = (𝑦 ∈ βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛) ↦ ⟨dom 𝑦, ((π‘†β€˜dom 𝑦)β€˜π‘¦)⟩)
pwfseqlem5.i 𝐼 = (π‘₯ ∈ Ο‰, 𝑦 ∈ 𝑑 ↦ ⟨(π‘‚β€˜π‘₯), π‘¦βŸ©)
pwfseqlem5.k 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
Assertion
Ref Expression
pwfseqlem5 Β¬ πœ‘
Distinct variable groups:   𝑛,𝑏,𝐺   π‘Ÿ,𝑏,𝑑,𝐻   𝑓,π‘˜,π‘₯,𝑃   𝑓,𝑏,π‘˜,𝑒,𝑣,π‘₯,𝑦,𝑛,π‘Ÿ,𝑑   πœ‘,𝑏,π‘˜,𝑛,π‘Ÿ,𝑑,π‘₯,𝑦   𝐾,𝑏,𝑛   𝑁,𝑏   πœ“,π‘˜,𝑛,π‘₯,𝑦   𝑆,𝑛,𝑦   𝐴,𝑏,𝑛,π‘Ÿ,𝑑   𝑂,𝑏,𝑒,𝑣,π‘₯,𝑦
Allowed substitution hints:   πœ‘(𝑣,𝑒,𝑓)   πœ“(𝑣,𝑒,𝑑,𝑓,π‘Ÿ,𝑏)   𝐴(π‘₯,𝑦,𝑣,𝑒,𝑓,π‘˜)   𝑃(𝑦,𝑣,𝑒,𝑑,𝑛,π‘Ÿ,𝑏)   𝑄(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,𝑛,π‘Ÿ,𝑏)   𝑆(π‘₯,𝑣,𝑒,𝑑,𝑓,π‘˜,π‘Ÿ,𝑏)   𝑇(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,𝑛,π‘Ÿ,𝑏)   𝐺(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,π‘Ÿ)   𝐻(π‘₯,𝑦,𝑣,𝑒,𝑓,π‘˜,𝑛)   𝐼(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,𝑛,π‘Ÿ,𝑏)   𝐾(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,π‘Ÿ)   𝑁(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,𝑛,π‘Ÿ)   𝑂(𝑑,𝑓,π‘˜,𝑛,π‘Ÿ)   𝑋(π‘₯,𝑦,𝑣,𝑒,𝑑,𝑓,π‘˜,𝑛,π‘Ÿ,𝑏)

Proof of Theorem pwfseqlem5
Dummy variables π‘Ž 𝑐 𝑑 𝑖 𝑗 π‘š 𝑠 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwfseqlem5.g . 2 (πœ‘ β†’ 𝐺:𝒫 𝐴–1-1β†’βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
2 pwfseqlem5.x . 2 (πœ‘ β†’ 𝑋 βŠ† 𝐴)
3 pwfseqlem5.h . 2 (πœ‘ β†’ 𝐻:ω–1-1-onto→𝑋)
4 pwfseqlem5.ps . 2 (πœ“ ↔ ((𝑑 βŠ† 𝐴 ∧ π‘Ÿ βŠ† (𝑑 Γ— 𝑑) ∧ π‘Ÿ We 𝑑) ∧ Ο‰ β‰Ό 𝑑))
5 vex 3448 . . . . . . . . . . 11 𝑑 ∈ V
6 simprl3 1221 . . . . . . . . . . . 12 ((πœ‘ ∧ ((𝑑 βŠ† 𝐴 ∧ π‘Ÿ βŠ† (𝑑 Γ— 𝑑) ∧ π‘Ÿ We 𝑑) ∧ Ο‰ β‰Ό 𝑑)) β†’ π‘Ÿ We 𝑑)
74, 6sylan2b 595 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ π‘Ÿ We 𝑑)
8 pwfseqlem5.o . . . . . . . . . . . 12 𝑂 = OrdIso(π‘Ÿ, 𝑑)
98oiiso 9478 . . . . . . . . . . 11 ((𝑑 ∈ V ∧ π‘Ÿ We 𝑑) β†’ 𝑂 Isom E , π‘Ÿ (dom 𝑂, 𝑑))
105, 7, 9sylancr 588 . . . . . . . . . 10 ((πœ‘ ∧ πœ“) β†’ 𝑂 Isom E , π‘Ÿ (dom 𝑂, 𝑑))
11 isof1o 7269 . . . . . . . . . 10 (𝑂 Isom E , π‘Ÿ (dom 𝑂, 𝑑) β†’ 𝑂:dom 𝑂–1-1-onto→𝑑)
1210, 11syl 17 . . . . . . . . 9 ((πœ‘ ∧ πœ“) β†’ 𝑂:dom 𝑂–1-1-onto→𝑑)
13 cardom 9927 . . . . . . . . . . . 12 (cardβ€˜Ο‰) = Ο‰
14 simprr 772 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ ((𝑑 βŠ† 𝐴 ∧ π‘Ÿ βŠ† (𝑑 Γ— 𝑑) ∧ π‘Ÿ We 𝑑) ∧ Ο‰ β‰Ό 𝑑)) β†’ Ο‰ β‰Ό 𝑑)
154, 14sylan2b 595 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ Ο‰ β‰Ό 𝑑)
168oien 9479 . . . . . . . . . . . . . . . 16 ((𝑑 ∈ V ∧ π‘Ÿ We 𝑑) β†’ dom 𝑂 β‰ˆ 𝑑)
175, 7, 16sylancr 588 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ πœ“) β†’ dom 𝑂 β‰ˆ 𝑑)
1817ensymd 8948 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ 𝑑 β‰ˆ dom 𝑂)
19 domentr 8956 . . . . . . . . . . . . . 14 ((Ο‰ β‰Ό 𝑑 ∧ 𝑑 β‰ˆ dom 𝑂) β†’ Ο‰ β‰Ό dom 𝑂)
2015, 18, 19syl2anc 585 . . . . . . . . . . . . 13 ((πœ‘ ∧ πœ“) β†’ Ο‰ β‰Ό dom 𝑂)
21 omelon 9587 . . . . . . . . . . . . . . 15 Ο‰ ∈ On
22 onenon 9890 . . . . . . . . . . . . . . 15 (Ο‰ ∈ On β†’ Ο‰ ∈ dom card)
2321, 22ax-mp 5 . . . . . . . . . . . . . 14 Ο‰ ∈ dom card
248oion 9477 . . . . . . . . . . . . . . . 16 (𝑑 ∈ V β†’ dom 𝑂 ∈ On)
2524elv 3450 . . . . . . . . . . . . . . 15 dom 𝑂 ∈ On
26 onenon 9890 . . . . . . . . . . . . . . 15 (dom 𝑂 ∈ On β†’ dom 𝑂 ∈ dom card)
2725, 26mp1i 13 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ dom 𝑂 ∈ dom card)
28 carddom2 9918 . . . . . . . . . . . . . 14 ((Ο‰ ∈ dom card ∧ dom 𝑂 ∈ dom card) β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜dom 𝑂) ↔ Ο‰ β‰Ό dom 𝑂))
2923, 27, 28sylancr 588 . . . . . . . . . . . . 13 ((πœ‘ ∧ πœ“) β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜dom 𝑂) ↔ Ο‰ β‰Ό dom 𝑂))
3020, 29mpbird 257 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ (cardβ€˜Ο‰) βŠ† (cardβ€˜dom 𝑂))
3113, 30eqsstrrid 3994 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ Ο‰ βŠ† (cardβ€˜dom 𝑂))
32 cardonle 9898 . . . . . . . . . . . 12 (dom 𝑂 ∈ On β†’ (cardβ€˜dom 𝑂) βŠ† dom 𝑂)
3325, 32mp1i 13 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ (cardβ€˜dom 𝑂) βŠ† dom 𝑂)
3431, 33sstrd 3955 . . . . . . . . . 10 ((πœ‘ ∧ πœ“) β†’ Ο‰ βŠ† dom 𝑂)
35 sseq2 3971 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 β†’ (Ο‰ βŠ† 𝑏 ↔ Ο‰ βŠ† dom 𝑂))
36 fveq2 6843 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 β†’ (π‘β€˜π‘) = (π‘β€˜dom 𝑂))
3736f1oeq1d 6780 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 β†’ ((π‘β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏 ↔ (π‘β€˜dom 𝑂):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
38 xpeq12 5659 . . . . . . . . . . . . . . 15 ((𝑏 = dom 𝑂 ∧ 𝑏 = dom 𝑂) β†’ (𝑏 Γ— 𝑏) = (dom 𝑂 Γ— dom 𝑂))
3938anidms 568 . . . . . . . . . . . . . 14 (𝑏 = dom 𝑂 β†’ (𝑏 Γ— 𝑏) = (dom 𝑂 Γ— dom 𝑂))
4039f1oeq2d 6781 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 β†’ ((π‘β€˜dom 𝑂):(𝑏 Γ— 𝑏)–1-1-onto→𝑏 ↔ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-onto→𝑏))
41 f1oeq3 6775 . . . . . . . . . . . . 13 (𝑏 = dom 𝑂 β†’ ((π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-onto→𝑏 ↔ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’dom 𝑂))
4237, 40, 413bitrd 305 . . . . . . . . . . . 12 (𝑏 = dom 𝑂 β†’ ((π‘β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏 ↔ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’dom 𝑂))
4335, 42imbi12d 345 . . . . . . . . . . 11 (𝑏 = dom 𝑂 β†’ ((Ο‰ βŠ† 𝑏 β†’ (π‘β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏) ↔ (Ο‰ βŠ† dom 𝑂 β†’ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’dom 𝑂)))
44 pwfseqlem5.n . . . . . . . . . . . 12 (πœ‘ β†’ βˆ€π‘ ∈ (harβ€˜π’« 𝐴)(Ο‰ βŠ† 𝑏 β†’ (π‘β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
4544adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ βˆ€π‘ ∈ (harβ€˜π’« 𝐴)(Ο‰ βŠ† 𝑏 β†’ (π‘β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
4625a1i 11 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ dom 𝑂 ∈ On)
471adantr 482 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ πœ“) β†’ 𝐺:𝒫 𝐴–1-1β†’βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
48 omex 9584 . . . . . . . . . . . . . . . . . 18 Ο‰ ∈ V
49 ovex 7391 . . . . . . . . . . . . . . . . . 18 (𝐴 ↑m 𝑛) ∈ V
5048, 49iunex 7902 . . . . . . . . . . . . . . . . 17 βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∈ V
51 f1dmex 7890 . . . . . . . . . . . . . . . . 17 ((𝐺:𝒫 𝐴–1-1β†’βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∈ V) β†’ 𝒫 𝐴 ∈ V)
5247, 50, 51sylancl 587 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ πœ“) β†’ 𝒫 𝐴 ∈ V)
53 pwexb 7701 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
5452, 53sylibr 233 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ πœ“) β†’ 𝐴 ∈ V)
55 simprl1 1219 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ ((𝑑 βŠ† 𝐴 ∧ π‘Ÿ βŠ† (𝑑 Γ— 𝑑) ∧ π‘Ÿ We 𝑑) ∧ Ο‰ β‰Ό 𝑑)) β†’ 𝑑 βŠ† 𝐴)
564, 55sylan2b 595 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ πœ“) β†’ 𝑑 βŠ† 𝐴)
57 ssdomg 8943 . . . . . . . . . . . . . . 15 (𝐴 ∈ V β†’ (𝑑 βŠ† 𝐴 β†’ 𝑑 β‰Ό 𝐴))
5854, 56, 57sylc 65 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ 𝑑 β‰Ό 𝐴)
59 canth2g 9078 . . . . . . . . . . . . . . 15 (𝐴 ∈ V β†’ 𝐴 β‰Ί 𝒫 𝐴)
60 sdomdom 8923 . . . . . . . . . . . . . . 15 (𝐴 β‰Ί 𝒫 𝐴 β†’ 𝐴 β‰Ό 𝒫 𝐴)
6154, 59, 603syl 18 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ 𝐴 β‰Ό 𝒫 𝐴)
62 domtr 8950 . . . . . . . . . . . . . 14 ((𝑑 β‰Ό 𝐴 ∧ 𝐴 β‰Ό 𝒫 𝐴) β†’ 𝑑 β‰Ό 𝒫 𝐴)
6358, 61, 62syl2anc 585 . . . . . . . . . . . . 13 ((πœ‘ ∧ πœ“) β†’ 𝑑 β‰Ό 𝒫 𝐴)
64 endomtr 8955 . . . . . . . . . . . . 13 ((dom 𝑂 β‰ˆ 𝑑 ∧ 𝑑 β‰Ό 𝒫 𝐴) β†’ dom 𝑂 β‰Ό 𝒫 𝐴)
6517, 63, 64syl2anc 585 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ dom 𝑂 β‰Ό 𝒫 𝐴)
66 elharval 9502 . . . . . . . . . . . 12 (dom 𝑂 ∈ (harβ€˜π’« 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 β‰Ό 𝒫 𝐴))
6746, 65, 66sylanbrc 584 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ dom 𝑂 ∈ (harβ€˜π’« 𝐴))
6843, 45, 67rspcdva 3581 . . . . . . . . . 10 ((πœ‘ ∧ πœ“) β†’ (Ο‰ βŠ† dom 𝑂 β†’ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’dom 𝑂))
6934, 68mpd 15 . . . . . . . . 9 ((πœ‘ ∧ πœ“) β†’ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’dom 𝑂)
70 f1oco 6808 . . . . . . . . 9 ((𝑂:dom 𝑂–1-1-onto→𝑑 ∧ (π‘β€˜dom 𝑂):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’dom 𝑂) β†’ (𝑂 ∘ (π‘β€˜dom 𝑂)):(dom 𝑂 Γ— dom 𝑂)–1-1-onto→𝑑)
7112, 69, 70syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ πœ“) β†’ (𝑂 ∘ (π‘β€˜dom 𝑂)):(dom 𝑂 Γ— dom 𝑂)–1-1-onto→𝑑)
72 f1of 6785 . . . . . . . . . . . . . . 15 (𝑂:dom 𝑂–1-1-onto→𝑑 β†’ 𝑂:dom π‘‚βŸΆπ‘‘)
7312, 72syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ 𝑂:dom π‘‚βŸΆπ‘‘)
7473feqmptd 6911 . . . . . . . . . . . . 13 ((πœ‘ ∧ πœ“) β†’ 𝑂 = (𝑒 ∈ dom 𝑂 ↦ (π‘‚β€˜π‘’)))
7574f1oeq1d 6780 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ (𝑂:dom 𝑂–1-1-onto→𝑑 ↔ (𝑒 ∈ dom 𝑂 ↦ (π‘‚β€˜π‘’)):dom 𝑂–1-1-onto→𝑑))
7612, 75mpbid 231 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ (𝑒 ∈ dom 𝑂 ↦ (π‘‚β€˜π‘’)):dom 𝑂–1-1-onto→𝑑)
7773feqmptd 6911 . . . . . . . . . . . . 13 ((πœ‘ ∧ πœ“) β†’ 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (π‘‚β€˜π‘£)))
7877f1oeq1d 6780 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ (𝑂:dom 𝑂–1-1-onto→𝑑 ↔ (𝑣 ∈ dom 𝑂 ↦ (π‘‚β€˜π‘£)):dom 𝑂–1-1-onto→𝑑))
7912, 78mpbid 231 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ (𝑣 ∈ dom 𝑂 ↦ (π‘‚β€˜π‘£)):dom 𝑂–1-1-onto→𝑑)
8076, 79xpf1o 9086 . . . . . . . . . 10 ((πœ‘ ∧ πœ“) β†’ (𝑒 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(π‘‚β€˜π‘’), (π‘‚β€˜π‘£)⟩):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑))
81 pwfseqlem5.t . . . . . . . . . . 11 𝑇 = (𝑒 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(π‘‚β€˜π‘’), (π‘‚β€˜π‘£)⟩)
82 f1oeq1 6773 . . . . . . . . . . 11 (𝑇 = (𝑒 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(π‘‚β€˜π‘’), (π‘‚β€˜π‘£)⟩) β†’ (𝑇:(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑) ↔ (𝑒 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(π‘‚β€˜π‘’), (π‘‚β€˜π‘£)⟩):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑)))
8381, 82ax-mp 5 . . . . . . . . . 10 (𝑇:(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑) ↔ (𝑒 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(π‘‚β€˜π‘’), (π‘‚β€˜π‘£)⟩):(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑))
8480, 83sylibr 233 . . . . . . . . 9 ((πœ‘ ∧ πœ“) β†’ 𝑇:(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑))
85 f1ocnv 6797 . . . . . . . . 9 (𝑇:(dom 𝑂 Γ— dom 𝑂)–1-1-ontoβ†’(𝑑 Γ— 𝑑) β†’ ◑𝑇:(𝑑 Γ— 𝑑)–1-1-ontoβ†’(dom 𝑂 Γ— dom 𝑂))
8684, 85syl 17 . . . . . . . 8 ((πœ‘ ∧ πœ“) β†’ ◑𝑇:(𝑑 Γ— 𝑑)–1-1-ontoβ†’(dom 𝑂 Γ— dom 𝑂))
87 f1oco 6808 . . . . . . . 8 (((𝑂 ∘ (π‘β€˜dom 𝑂)):(dom 𝑂 Γ— dom 𝑂)–1-1-onto→𝑑 ∧ ◑𝑇:(𝑑 Γ— 𝑑)–1-1-ontoβ†’(dom 𝑂 Γ— dom 𝑂)) β†’ ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇):(𝑑 Γ— 𝑑)–1-1-onto→𝑑)
8871, 86, 87syl2anc 585 . . . . . . 7 ((πœ‘ ∧ πœ“) β†’ ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇):(𝑑 Γ— 𝑑)–1-1-onto→𝑑)
89 pwfseqlem5.p . . . . . . . 8 𝑃 = ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇)
90 f1oeq1 6773 . . . . . . . 8 (𝑃 = ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇) β†’ (𝑃:(𝑑 Γ— 𝑑)–1-1-onto→𝑑 ↔ ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇):(𝑑 Γ— 𝑑)–1-1-onto→𝑑))
9189, 90ax-mp 5 . . . . . . 7 (𝑃:(𝑑 Γ— 𝑑)–1-1-onto→𝑑 ↔ ((𝑂 ∘ (π‘β€˜dom 𝑂)) ∘ ◑𝑇):(𝑑 Γ— 𝑑)–1-1-onto→𝑑)
9288, 91sylibr 233 . . . . . 6 ((πœ‘ ∧ πœ“) β†’ 𝑃:(𝑑 Γ— 𝑑)–1-1-onto→𝑑)
93 f1of1 6784 . . . . . 6 (𝑃:(𝑑 Γ— 𝑑)–1-1-onto→𝑑 β†’ 𝑃:(𝑑 Γ— 𝑑)–1-1→𝑑)
9492, 93syl 17 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝑃:(𝑑 Γ— 𝑑)–1-1→𝑑)
95 f1of1 6784 . . . . . . . . . . . . 13 (𝑂:dom 𝑂–1-1-onto→𝑑 β†’ 𝑂:dom 𝑂–1-1→𝑑)
9612, 95syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ 𝑂:dom 𝑂–1-1→𝑑)
97 f1ssres 6747 . . . . . . . . . . . 12 ((𝑂:dom 𝑂–1-1→𝑑 ∧ Ο‰ βŠ† dom 𝑂) β†’ (𝑂 β†Ύ Ο‰):ω–1-1→𝑑)
9896, 34, 97syl2anc 585 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ (𝑂 β†Ύ Ο‰):ω–1-1→𝑑)
99 f1f1orn 6796 . . . . . . . . . . 11 ((𝑂 β†Ύ Ο‰):ω–1-1→𝑑 β†’ (𝑂 β†Ύ Ο‰):ω–1-1-ontoβ†’ran (𝑂 β†Ύ Ο‰))
10098, 99syl 17 . . . . . . . . . 10 ((πœ‘ ∧ πœ“) β†’ (𝑂 β†Ύ Ο‰):ω–1-1-ontoβ†’ran (𝑂 β†Ύ Ο‰))
10173, 34feqresmpt 6912 . . . . . . . . . . 11 ((πœ‘ ∧ πœ“) β†’ (𝑂 β†Ύ Ο‰) = (π‘₯ ∈ Ο‰ ↦ (π‘‚β€˜π‘₯)))
102101f1oeq1d 6780 . . . . . . . . . 10 ((πœ‘ ∧ πœ“) β†’ ((𝑂 β†Ύ Ο‰):ω–1-1-ontoβ†’ran (𝑂 β†Ύ Ο‰) ↔ (π‘₯ ∈ Ο‰ ↦ (π‘‚β€˜π‘₯)):ω–1-1-ontoβ†’ran (𝑂 β†Ύ Ο‰)))
103100, 102mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ πœ“) β†’ (π‘₯ ∈ Ο‰ ↦ (π‘‚β€˜π‘₯)):ω–1-1-ontoβ†’ran (𝑂 β†Ύ Ο‰))
104 mptresid 6005 . . . . . . . . . . 11 ( I β†Ύ 𝑑) = (𝑦 ∈ 𝑑 ↦ 𝑦)
105104eqcomi 2742 . . . . . . . . . 10 (𝑦 ∈ 𝑑 ↦ 𝑦) = ( I β†Ύ 𝑑)
106 f1oi 6823 . . . . . . . . . . 11 ( I β†Ύ 𝑑):𝑑–1-1-onto→𝑑
107 f1oeq1 6773 . . . . . . . . . . 11 ((𝑦 ∈ 𝑑 ↦ 𝑦) = ( I β†Ύ 𝑑) β†’ ((𝑦 ∈ 𝑑 ↦ 𝑦):𝑑–1-1-onto→𝑑 ↔ ( I β†Ύ 𝑑):𝑑–1-1-onto→𝑑))
108106, 107mpbiri 258 . . . . . . . . . 10 ((𝑦 ∈ 𝑑 ↦ 𝑦) = ( I β†Ύ 𝑑) β†’ (𝑦 ∈ 𝑑 ↦ 𝑦):𝑑–1-1-onto→𝑑)
109105, 108mp1i 13 . . . . . . . . 9 ((πœ‘ ∧ πœ“) β†’ (𝑦 ∈ 𝑑 ↦ 𝑦):𝑑–1-1-onto→𝑑)
110103, 109xpf1o 9086 . . . . . . . 8 ((πœ‘ ∧ πœ“) β†’ (π‘₯ ∈ Ο‰, 𝑦 ∈ 𝑑 ↦ ⟨(π‘‚β€˜π‘₯), π‘¦βŸ©):(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑))
111 pwfseqlem5.i . . . . . . . . 9 𝐼 = (π‘₯ ∈ Ο‰, 𝑦 ∈ 𝑑 ↦ ⟨(π‘‚β€˜π‘₯), π‘¦βŸ©)
112 f1oeq1 6773 . . . . . . . . 9 (𝐼 = (π‘₯ ∈ Ο‰, 𝑦 ∈ 𝑑 ↦ ⟨(π‘‚β€˜π‘₯), π‘¦βŸ©) β†’ (𝐼:(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) ↔ (π‘₯ ∈ Ο‰, 𝑦 ∈ 𝑑 ↦ ⟨(π‘‚β€˜π‘₯), π‘¦βŸ©):(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑)))
113111, 112ax-mp 5 . . . . . . . 8 (𝐼:(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) ↔ (π‘₯ ∈ Ο‰, 𝑦 ∈ 𝑑 ↦ ⟨(π‘‚β€˜π‘₯), π‘¦βŸ©):(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑))
114110, 113sylibr 233 . . . . . . 7 ((πœ‘ ∧ πœ“) β†’ 𝐼:(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑))
115 f1of1 6784 . . . . . . 7 (𝐼:(Ο‰ Γ— 𝑑)–1-1-ontoβ†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) β†’ 𝐼:(Ο‰ Γ— 𝑑)–1-1β†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑))
116114, 115syl 17 . . . . . 6 ((πœ‘ ∧ πœ“) β†’ 𝐼:(Ο‰ Γ— 𝑑)–1-1β†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑))
117 f1f 6739 . . . . . . 7 ((𝑂 β†Ύ Ο‰):ω–1-1→𝑑 β†’ (𝑂 β†Ύ Ο‰):Ο‰βŸΆπ‘‘)
118 frn 6676 . . . . . . 7 ((𝑂 β†Ύ Ο‰):Ο‰βŸΆπ‘‘ β†’ ran (𝑂 β†Ύ Ο‰) βŠ† 𝑑)
119 xpss1 5653 . . . . . . 7 (ran (𝑂 β†Ύ Ο‰) βŠ† 𝑑 β†’ (ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) βŠ† (𝑑 Γ— 𝑑))
12098, 117, 118, 1194syl 19 . . . . . 6 ((πœ‘ ∧ πœ“) β†’ (ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) βŠ† (𝑑 Γ— 𝑑))
121 f1ss 6745 . . . . . 6 ((𝐼:(Ο‰ Γ— 𝑑)–1-1β†’(ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) ∧ (ran (𝑂 β†Ύ Ο‰) Γ— 𝑑) βŠ† (𝑑 Γ— 𝑑)) β†’ 𝐼:(Ο‰ Γ— 𝑑)–1-1β†’(𝑑 Γ— 𝑑))
122116, 120, 121syl2anc 585 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝐼:(Ο‰ Γ— 𝑑)–1-1β†’(𝑑 Γ— 𝑑))
123 f1co 6751 . . . . 5 ((𝑃:(𝑑 Γ— 𝑑)–1-1→𝑑 ∧ 𝐼:(Ο‰ Γ— 𝑑)–1-1β†’(𝑑 Γ— 𝑑)) β†’ (𝑃 ∘ 𝐼):(Ο‰ Γ— 𝑑)–1-1→𝑑)
12494, 122, 123syl2anc 585 . . . 4 ((πœ‘ ∧ πœ“) β†’ (𝑃 ∘ 𝐼):(Ο‰ Γ— 𝑑)–1-1→𝑑)
1255a1i 11 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝑑 ∈ V)
126 peano1 7826 . . . . . . . 8 βˆ… ∈ Ο‰
127126a1i 11 . . . . . . 7 ((πœ‘ ∧ πœ“) β†’ βˆ… ∈ Ο‰)
12834, 127sseldd 3946 . . . . . 6 ((πœ‘ ∧ πœ“) β†’ βˆ… ∈ dom 𝑂)
12973, 128ffvelcdmd 7037 . . . . 5 ((πœ‘ ∧ πœ“) β†’ (π‘‚β€˜βˆ…) ∈ 𝑑)
130 pwfseqlem5.s . . . . 5 𝑆 = seqΟ‰((π‘˜ ∈ V, 𝑓 ∈ V ↦ (π‘₯ ∈ (𝑑 ↑m suc π‘˜) ↦ ((π‘“β€˜(π‘₯ β†Ύ π‘˜))𝑃(π‘₯β€˜π‘˜)))), {βŸ¨βˆ…, (π‘‚β€˜βˆ…)⟩})
131 pwfseqlem5.q . . . . 5 𝑄 = (𝑦 ∈ βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛) ↦ ⟨dom 𝑦, ((π‘†β€˜dom 𝑦)β€˜π‘¦)⟩)
132125, 129, 92, 130, 131fseqenlem2 9966 . . . 4 ((πœ‘ ∧ πœ“) β†’ 𝑄:βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1β†’(Ο‰ Γ— 𝑑))
133 f1co 6751 . . . 4 (((𝑃 ∘ 𝐼):(Ο‰ Γ— 𝑑)–1-1→𝑑 ∧ 𝑄:βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1β†’(Ο‰ Γ— 𝑑)) β†’ ((𝑃 ∘ 𝐼) ∘ 𝑄):βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑)
134124, 132, 133syl2anc 585 . . 3 ((πœ‘ ∧ πœ“) β†’ ((𝑃 ∘ 𝐼) ∘ 𝑄):βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑)
135 pwfseqlem5.k . . . 4 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
136 f1eq1 6734 . . . 4 (𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄) β†’ (𝐾:βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑))
137135, 136ax-mp 5 . . 3 (𝐾:βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑)
138134, 137sylibr 233 . 2 ((πœ‘ ∧ πœ“) β†’ 𝐾:βˆͺ 𝑛 ∈ Ο‰ (𝑑 ↑m 𝑛)–1-1→𝑑)
139 eqid 2733 . 2 (πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))}) = (πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})
140 eqid 2733 . 2 (𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑}))) = (𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))
141 eqid 2733 . . 3 {βŸ¨π‘, π‘‘βŸ© ∣ ((𝑐 βŠ† 𝐴 ∧ 𝑑 βŠ† (𝑐 Γ— 𝑐)) ∧ (𝑑 We 𝑐 ∧ βˆ€π‘š ∈ 𝑐 [(◑𝑑 β€œ {π‘š}) / 𝑗](𝑗(𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))(𝑑 ∩ (𝑗 Γ— 𝑗))) = π‘š))} = {βŸ¨π‘, π‘‘βŸ© ∣ ((𝑐 βŠ† 𝐴 ∧ 𝑑 βŠ† (𝑐 Γ— 𝑐)) ∧ (𝑑 We 𝑐 ∧ βˆ€π‘š ∈ 𝑐 [(◑𝑑 β€œ {π‘š}) / 𝑗](𝑗(𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))(𝑑 ∩ (𝑗 Γ— 𝑗))) = π‘š))}
142141fpwwe2cbv 10571 . 2 {βŸ¨π‘, π‘‘βŸ© ∣ ((𝑐 βŠ† 𝐴 ∧ 𝑑 βŠ† (𝑐 Γ— 𝑐)) ∧ (𝑑 We 𝑐 ∧ βˆ€π‘š ∈ 𝑐 [(◑𝑑 β€œ {π‘š}) / 𝑗](𝑗(𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))(𝑑 ∩ (𝑗 Γ— 𝑗))) = π‘š))} = {βŸ¨π‘Ž, π‘ βŸ© ∣ ((π‘Ž βŠ† 𝐴 ∧ 𝑠 βŠ† (π‘Ž Γ— π‘Ž)) ∧ (𝑠 We π‘Ž ∧ βˆ€π‘ ∈ π‘Ž [(◑𝑠 β€œ {𝑏}) / 𝑀](𝑀(𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))(𝑠 ∩ (𝑀 Γ— 𝑀))) = 𝑏))}
143 eqid 2733 . 2 βˆͺ dom {βŸ¨π‘, π‘‘βŸ© ∣ ((𝑐 βŠ† 𝐴 ∧ 𝑑 βŠ† (𝑐 Γ— 𝑐)) ∧ (𝑑 We 𝑐 ∧ βˆ€π‘š ∈ 𝑐 [(◑𝑑 β€œ {π‘š}) / 𝑗](𝑗(𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))(𝑑 ∩ (𝑗 Γ— 𝑗))) = π‘š))} = βˆͺ dom {βŸ¨π‘, π‘‘βŸ© ∣ ((𝑐 βŠ† 𝐴 ∧ 𝑑 βŠ† (𝑐 Γ— 𝑐)) ∧ (𝑑 We 𝑐 ∧ βˆ€π‘š ∈ 𝑐 [(◑𝑑 β€œ {π‘š}) / 𝑗](𝑗(𝑑 ∈ V, π‘Ÿ ∈ V ↦ if(𝑑 ∈ Fin, (π»β€˜(cardβ€˜π‘‘)), ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ ((πΊβ€˜{𝑖 ∈ 𝑑 ∣ ((β—‘πΎβ€˜π‘–) ∈ ran 𝐺 ∧ Β¬ 𝑖 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘–)))})β€˜π‘§) ∈ 𝑑})))(𝑑 ∩ (𝑗 Γ— 𝑗))) = π‘š))}
1441, 2, 3, 4, 138, 139, 140, 142, 143pwfseqlem4 10603 1 Β¬ πœ‘
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3444  [wsbc 3740   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487  π’« cpw 4561  {csn 4587  βŸ¨cop 4593  βˆͺ cuni 4866  βˆ© cint 4908  βˆͺ ciun 4955   class class class wbr 5106  {copab 5168   ↦ cmpt 5189   I cid 5531   E cep 5537   We wwe 5588   Γ— cxp 5632  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β†Ύ cres 5636   β€œ cima 5637   ∘ ccom 5638  Oncon0 6318  suc csuc 6320  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497   Isom wiso 6498  (class class class)co 7358   ∈ cmpo 7360  Ο‰com 7803  seqΟ‰cseqom 8394   ↑m cmap 8768   β‰ˆ cen 8883   β‰Ό cdom 8884   β‰Ί csdm 8885  Fincfn 8886  OrdIsocoi 9450  harchar 9497  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seqom 8395  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-oi 9451  df-har 9498  df-card 9880
This theorem is referenced by:  pwfseq  10605
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