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Theorem pwfseqlem4 9428
Description: Lemma for pwfseq 9430. Derive a final contradiction from the function 𝐹 in pwfseqlem3 9426. Applying fpwwe2 9409 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊𝑍)) ∈ 𝑍 contradicts our assumption on 𝐹, so we are reduced to the case of 𝑍 finite. This too is a contradiction, though, because 𝑍 and its preimage under (𝑊𝑍) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
Hypotheses
Ref Expression
pwfseqlem4.g (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
pwfseqlem4.x (𝜑𝑋𝐴)
pwfseqlem4.h (𝜑𝐻:ω–1-1-onto𝑋)
pwfseqlem4.ps (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
pwfseqlem4.d 𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
pwfseqlem4.f 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
pwfseqlem4.w 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏𝑎 [(𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}
pwfseqlem4.z 𝑍 = dom 𝑊
Assertion
Ref Expression
pwfseqlem4 ¬ 𝜑
Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑎,𝑏,𝑠,𝑣,𝐹   𝑤,𝐺   𝑤,𝐾   𝑟,𝑎,𝑥,𝑧,𝐻,𝑏,𝑠,𝑣   𝑛,𝑎,𝜑,𝑏,𝑠,𝑣,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑎,𝑛,𝑟,𝑠,𝑥,𝑧   𝑊,𝑎,𝑏,𝑠,𝑣   𝑍,𝑎,𝑏,𝑠,𝑣
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑣,𝑠,𝑟,𝑎,𝑏)   𝐴(𝑤,𝑣,𝑏)   𝐷(𝑥,𝑤,𝑣,𝑠,𝑟,𝑎,𝑏)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑧,𝑤,𝑛,𝑟)   𝑋(𝑥,𝑧,𝑤,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝑍(𝑥,𝑧,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem4
StepHypRef Expression
1 eqid 2621 . . . . . . . . . . 11 𝑍 = 𝑍
2 eqid 2621 . . . . . . . . . . 11 (𝑊𝑍) = (𝑊𝑍)
31, 2pm3.2i 471 . . . . . . . . . 10 (𝑍 = 𝑍 ∧ (𝑊𝑍) = (𝑊𝑍))
4 pwfseqlem4.w . . . . . . . . . . 11 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏𝑎 [(𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}
5 pwfseqlem4.g . . . . . . . . . . . . 13 (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
6 omex 8484 . . . . . . . . . . . . . 14 ω ∈ V
7 ovex 6632 . . . . . . . . . . . . . 14 (𝐴𝑚 𝑛) ∈ V
86, 7iunex 7093 . . . . . . . . . . . . 13 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
9 f1dmex 7083 . . . . . . . . . . . . 13 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
105, 8, 9sylancl 693 . . . . . . . . . . . 12 (𝜑 → 𝒫 𝐴 ∈ V)
11 pwexb 6922 . . . . . . . . . . . 12 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
1210, 11sylibr 224 . . . . . . . . . . 11 (𝜑𝐴 ∈ V)
13 pwfseqlem4.x . . . . . . . . . . . 12 (𝜑𝑋𝐴)
14 pwfseqlem4.h . . . . . . . . . . . 12 (𝜑𝐻:ω–1-1-onto𝑋)
15 pwfseqlem4.ps . . . . . . . . . . . 12 (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
16 pwfseqlem4.k . . . . . . . . . . . 12 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
17 pwfseqlem4.d . . . . . . . . . . . 12 𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
18 pwfseqlem4.f . . . . . . . . . . . 12 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
195, 13, 14, 15, 16, 17, 18pwfseqlem4a 9427 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20 pwfseqlem4.z . . . . . . . . . . 11 𝑍 = dom 𝑊
214, 12, 19, 20fpwwe2 9409 . . . . . . . . . 10 (𝜑 → ((𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊𝑍) = (𝑊𝑍))))
223, 21mpbiri 248 . . . . . . . . 9 (𝜑 → (𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
2322simprd 479 . . . . . . . 8 (𝜑 → (𝑍𝐹(𝑊𝑍)) ∈ 𝑍)
2422simpld 475 . . . . . . . . . . . . 13 (𝜑𝑍𝑊(𝑊𝑍))
254, 12fpwwe2lem2 9398 . . . . . . . . . . . . 13 (𝜑 → (𝑍𝑊(𝑊𝑍) ↔ ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))))
2624, 25mpbid 222 . . . . . . . . . . . 12 (𝜑 → ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)))
2726simpld 475 . . . . . . . . . . 11 (𝜑 → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
2827simpld 475 . . . . . . . . . 10 (𝜑𝑍𝐴)
2912, 28ssexd 4765 . . . . . . . . 9 (𝜑𝑍 ∈ V)
30 sseq1 3605 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → (𝑎𝐴𝑍𝐴))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑍𝑎 = 𝑍)
3231sqxpeqd 5101 . . . . . . . . . . . . . . 15 (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
3332sseq2d 3612 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
34 weeq2 5063 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) We 𝑎 ↔ (𝑊𝑍) We 𝑍))
3530, 33, 343anbi123d 1396 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → ((𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎) ↔ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
3635anbi2d 739 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) ↔ (𝜑 ∧ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))))
37 id 22 . . . . . . . . . . . . . . . 16 ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍) → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
38373expa 1262 . . . . . . . . . . . . . . 15 (((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊𝑍) We 𝑍) → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
3938adantrr 752 . . . . . . . . . . . . . 14 (((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)) → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
4026, 39syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
4140pm4.71i 663 . . . . . . . . . . . 12 (𝜑 ↔ (𝜑 ∧ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
4236, 41syl6bbr 278 . . . . . . . . . . 11 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) ↔ 𝜑))
43 oveq1 6611 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → (𝑎𝐹(𝑊𝑍)) = (𝑍𝐹(𝑊𝑍)))
4443, 31eleq12d 2692 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
45 breq1 4616 . . . . . . . . . . . 12 (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
4644, 45imbi12d 334 . . . . . . . . . . 11 (𝑎 = 𝑍 → (((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω)))
4742, 46imbi12d 334 . . . . . . . . . 10 (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω))))
48 fvex 6158 . . . . . . . . . . 11 (𝑊𝑍) ∈ V
49 sseq1 3605 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑎 × 𝑎)))
50 weeq1 5062 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 We 𝑎 ↔ (𝑊𝑍) We 𝑎))
5149, 503anbi23d 1399 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)))
5251anbi2d 739 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎))))
53 oveq2 6612 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊𝑍)))
5453eleq1d 2683 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊𝑍)) ∈ 𝑎))
5554imbi1d 331 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)))
5652, 55imbi12d 334 . . . . . . . . . . 11 (𝑠 = (𝑊𝑍) → (((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω)) ↔ ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω))))
57 omelon 8487 . . . . . . . . . . . . . . 15 ω ∈ On
58 onenon 8719 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
5957, 58ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
60 simpr3 1067 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
61 19.8a 2049 . . . . . . . . . . . . . . . 16 (𝑠 We 𝑎 → ∃𝑠 𝑠 We 𝑎)
6260, 61syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
63 ween 8802 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
6462, 63sylibr 224 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
65 domtri2 8759 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
6659, 64, 65sylancr 694 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
67 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
68 nfcv 2761 . . . . . . . . . . . . . . . . . . 19 𝑟𝑎
69 nfmpt22 6676 . . . . . . . . . . . . . . . . . . . 20 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
7018, 69nfcxfr 2759 . . . . . . . . . . . . . . . . . . 19 𝑟𝐹
71 nfcv 2761 . . . . . . . . . . . . . . . . . . 19 𝑟𝑠
7268, 70, 71nfov 6630 . . . . . . . . . . . . . . . . . 18 𝑟(𝑎𝐹𝑠)
7372nfel1 2775 . . . . . . . . . . . . . . . . 17 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
7467, 73nfim 1822 . . . . . . . . . . . . . . . 16 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
75 sseq1 3605 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
76 weeq1 5062 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
7775, 763anbi23d 1399 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
7877anbi1d 740 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
7978anbi2d 739 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
80 oveq2 6612 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
8180eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
8279, 81imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑠 → (((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))))
83 nfv 1840 . . . . . . . . . . . . . . . . . 18 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
84 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎
85 nfmpt21 6675 . . . . . . . . . . . . . . . . . . . . 21 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
8618, 85nfcxfr 2759 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
87 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑟
8884, 86, 87nfov 6630 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑎𝐹𝑟)
8988nfel1 2775 . . . . . . . . . . . . . . . . . 18 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
9083, 89nfim 1822 . . . . . . . . . . . . . . . . 17 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
91 sseq1 3605 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
92 xpeq12 5094 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9392anidms 676 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9493sseq2d 3612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
95 weeq2 5063 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9691, 94, 953anbi123d 1396 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
97 breq2 4617 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
9896, 97anbi12d 746 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9915, 98syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
10099anbi2d 739 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
101 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
102 difeq2 3700 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
103101, 102eleq12d 2692 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
104100, 103imbi12d 334 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
1055, 13, 14, 15, 16, 17, 18pwfseqlem3 9426 . . . . . . . . . . . . . . . . 17 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
10690, 104, 105chvar 2261 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
10774, 82, 106chvar 2261 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
108107eldifbd 3568 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎)
109108expr 642 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
11066, 109sylbird 250 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
111110con4d 114 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω))
11248, 56, 111vtocl 3245 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω))
11347, 112vtoclg 3252 . . . . . . . . 9 (𝑍 ∈ V → (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω)))
11429, 113mpcom 38 . . . . . . . 8 (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω))
11523, 114mpd 15 . . . . . . 7 (𝜑𝑍 ≺ ω)
116 isfinite 8493 . . . . . . 7 (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
117115, 116sylibr 224 . . . . . 6 (𝜑𝑍 ∈ Fin)
1185, 13, 14, 15, 16, 17, 18pwfseqlem2 9425 . . . . . 6 ((𝑍 ∈ Fin ∧ (𝑊𝑍) ∈ V) → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
119117, 48, 118sylancl 693 . . . . 5 (𝜑 → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
120119, 23eqeltrrd 2699 . . . 4 (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
1214, 12, 24fpwwe2lem3 9399 . . . . . . . . . 10 ((𝜑 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
122120, 121mpdan 701 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
123 cnvimass 5444 . . . . . . . . . . . 12 ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊𝑍)
12427simprd 479 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝑍) ⊆ (𝑍 × 𝑍))
125 dmss 5283 . . . . . . . . . . . . . 14 ((𝑊𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
126124, 125syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
127 dmxpss 5524 . . . . . . . . . . . . 13 dom (𝑍 × 𝑍) ⊆ 𝑍
128126, 127syl6ss 3595 . . . . . . . . . . . 12 (𝜑 → dom (𝑊𝑍) ⊆ 𝑍)
129123, 128syl5ss 3594 . . . . . . . . . . 11 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
130117, 129ssfid 8127 . . . . . . . . . 10 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
13148inex1 4759 . . . . . . . . . 10 ((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
1325, 13, 14, 15, 16, 17, 18pwfseqlem2 9425 . . . . . . . . . 10 ((((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin ∧ ((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V) → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
133130, 131, 132sylancl 693 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
134122, 133eqtr3d 2657 . . . . . . . 8 (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
135 f1of1 6093 . . . . . . . . . 10 (𝐻:ω–1-1-onto𝑋𝐻:ω–1-1𝑋)
13614, 135syl 17 . . . . . . . . 9 (𝜑𝐻:ω–1-1𝑋)
137 ficardom 8731 . . . . . . . . . 10 (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
138117, 137syl 17 . . . . . . . . 9 (𝜑 → (card‘𝑍) ∈ ω)
139 ficardom 8731 . . . . . . . . . 10 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
140130, 139syl 17 . . . . . . . . 9 (𝜑 → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
141 f1fveq 6473 . . . . . . . . 9 ((𝐻:ω–1-1𝑋 ∧ ((card‘𝑍) ∈ ω ∧ (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)) → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
142136, 138, 140, 141syl12anc 1321 . . . . . . . 8 (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
143134, 142mpbid 222 . . . . . . 7 (𝜑 → (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))
144143eqcomd 2627 . . . . . 6 (𝜑 → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
145 finnum 8718 . . . . . . . 8 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
146130, 145syl 17 . . . . . . 7 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
147 finnum 8718 . . . . . . . 8 (𝑍 ∈ Fin → 𝑍 ∈ dom card)
148117, 147syl 17 . . . . . . 7 (𝜑𝑍 ∈ dom card)
149 carden2 8757 . . . . . . 7 ((((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card ∧ 𝑍 ∈ dom card) → ((card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
150146, 148, 149syl2anc 692 . . . . . 6 (𝜑 → ((card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
151144, 150mpbid 222 . . . . 5 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
152 dfpss2 3670 . . . . . . . 8 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
153152baib 943 . . . . . . 7 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
154129, 153syl 17 . . . . . 6 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
155 php3 8090 . . . . . . . . 9 ((𝑍 ∈ Fin ∧ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
156 sdomnen 7928 . . . . . . . . 9 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
157155, 156syl 17 . . . . . . . 8 ((𝑍 ∈ Fin ∧ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
158157ex 450 . . . . . . 7 (𝑍 ∈ Fin → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
159117, 158syl 17 . . . . . 6 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
160154, 159sylbird 250 . . . . 5 (𝜑 → (¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
161151, 160mt4d 152 . . . 4 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)
162120, 161eleqtrrd 2701 . . 3 (𝜑 → (𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))
163 fvex 6158 . . . 4 (𝐻‘(card‘𝑍)) ∈ V
164163eliniseg 5453 . . . 4 ((𝐻‘(card‘𝑍)) ∈ V → ((𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍))))
165163, 164ax-mp 5 . . 3 ((𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
166162, 165sylib 208 . 2 (𝜑 → (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
16726simprd 479 . . . . 5 (𝜑 → ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))
168167simpld 475 . . . 4 (𝜑 → (𝑊𝑍) We 𝑍)
169 weso 5065 . . . 4 ((𝑊𝑍) We 𝑍 → (𝑊𝑍) Or 𝑍)
170168, 169syl 17 . . 3 (𝜑 → (𝑊𝑍) Or 𝑍)
171 sonr 5016 . . 3 (((𝑊𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
172170, 120, 171syl2anc 692 . 2 (𝜑 → ¬ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
173166, 172pm2.65i 185 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  [wsbc 3417  cdif 3552  cin 3554  wss 3555  wpss 3556  ifcif 4058  𝒫 cpw 4130  {csn 4148   cuni 4402   cint 4440   ciun 4485   class class class wbr 4613  {copab 4672   Or wor 4994   We wwe 5032   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  Oncon0 5682  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  𝑚 cmap 7802  cen 7896  cdom 7897  csdm 7898  Fincfn 7899  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-oi 8359  df-card 8709
This theorem is referenced by:  pwfseqlem5  9429
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