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Theorem pwfseqlem4 9737
Description: Lemma for pwfseq 9739. Derive a final contradiction from the function 𝐹 in pwfseqlem3 9735. Applying fpwwe2 9718 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 2765 . . . . . . . . . . 11 𝑍 = 𝑍
2 eqid 2765 . . . . . . . . . . 11 (𝑊𝑍) = (𝑊𝑍)
31, 2pm3.2i 462 . . . . . . . . . 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 8755 . . . . . . . . . . . . . 14 ω ∈ V
7 ovex 6874 . . . . . . . . . . . . . 14 (𝐴𝑚 𝑛) ∈ V
86, 7iunex 7345 . . . . . . . . . . . . 13 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
9 f1dmex 7334 . . . . . . . . . . . . 13 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
105, 8, 9sylancl 580 . . . . . . . . . . . 12 (𝜑 → 𝒫 𝐴 ∈ V)
11 pwexb 7173 . . . . . . . . . . . 12 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
1210, 11sylibr 225 . . . . . . . . . . 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 9736 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20 pwfseqlem4.z . . . . . . . . . . 11 𝑍 = dom 𝑊
214, 12, 19, 20fpwwe2 9718 . . . . . . . . . 10 (𝜑 → ((𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊𝑍) = (𝑊𝑍))))
223, 21mpbiri 249 . . . . . . . . 9 (𝜑 → (𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
2322simprd 489 . . . . . . . 8 (𝜑 → (𝑍𝐹(𝑊𝑍)) ∈ 𝑍)
2422simpld 488 . . . . . . . . . . . . 13 (𝜑𝑍𝑊(𝑊𝑍))
254, 12fpwwe2lem2 9707 . . . . . . . . . . . . 13 (𝜑 → (𝑍𝑊(𝑊𝑍) ↔ ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))))
2624, 25mpbid 223 . . . . . . . . . . . 12 (𝜑 → ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)))
2726simpld 488 . . . . . . . . . . 11 (𝜑 → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
2827simpld 488 . . . . . . . . . 10 (𝜑𝑍𝐴)
2912, 28ssexd 4966 . . . . . . . . 9 (𝜑𝑍 ∈ V)
30 sseq1 3786 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → (𝑎𝐴𝑍𝐴))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑍𝑎 = 𝑍)
3231sqxpeqd 5309 . . . . . . . . . . . . . . 15 (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
3332sseq2d 3793 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
34 weeq2 5266 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) We 𝑎 ↔ (𝑊𝑍) We 𝑍))
3530, 33, 343anbi123d 1560 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → ((𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎) ↔ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
3635anbi2d 622 . . . . . . . . . . . 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 1147 . . . . . . . . . . . . . . 15 (((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊𝑍) We 𝑍) → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
3938adantrr 708 . . . . . . . . . . . . . 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 555 . . . . . . . . . . . 12 (𝜑 ↔ (𝜑 ∧ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
4236, 41syl6bbr 280 . . . . . . . . . . 11 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) ↔ 𝜑))
43 oveq1 6849 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → (𝑎𝐹(𝑊𝑍)) = (𝑍𝐹(𝑊𝑍)))
4443, 31eleq12d 2838 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
45 breq1 4812 . . . . . . . . . . . 12 (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
4644, 45imbi12d 335 . . . . . . . . . . 11 (𝑎 = 𝑍 → (((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω)))
4742, 46imbi12d 335 . . . . . . . . . 10 (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω))))
48 fvex 6388 . . . . . . . . . . 11 (𝑊𝑍) ∈ V
49 sseq1 3786 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑎 × 𝑎)))
50 weeq1 5265 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 We 𝑎 ↔ (𝑊𝑍) We 𝑎))
5149, 503anbi23d 1563 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)))
5251anbi2d 622 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎))))
53 oveq2 6850 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊𝑍)))
5453eleq1d 2829 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊𝑍)) ∈ 𝑎))
5554imbi1d 332 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)))
5652, 55imbi12d 335 . . . . . . . . . . 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 8758 . . . . . . . . . . . . . . 15 ω ∈ On
58 onenon 9026 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
5957, 58ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
60 simpr3 1252 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
61 19.8a 2214 . . . . . . . . . . . . . . . 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 9109 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
6462, 63sylibr 225 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
65 domtri2 9066 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
6659, 64, 65sylancr 581 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
67 nfv 2009 . . . . . . . . . . . . . . . . 17 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
68 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑟𝑎
69 nfmpt22 6921 . . . . . . . . . . . . . . . . . . . 20 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
7018, 69nfcxfr 2905 . . . . . . . . . . . . . . . . . . 19 𝑟𝐹
71 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑟𝑠
7268, 70, 71nfov 6872 . . . . . . . . . . . . . . . . . 18 𝑟(𝑎𝐹𝑠)
7372nfel1 2922 . . . . . . . . . . . . . . . . 17 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
7467, 73nfim 1995 . . . . . . . . . . . . . . . 16 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
75 sseq1 3786 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
76 weeq1 5265 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
7775, 763anbi23d 1563 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
7877anbi1d 623 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
7978anbi2d 622 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
80 oveq2 6850 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
8180eleq1d 2829 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
8279, 81imbi12d 335 . . . . . . . . . . . . . . . 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 2009 . . . . . . . . . . . . . . . . . 18 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
84 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎
85 nfmpt21 6920 . . . . . . . . . . . . . . . . . . . . 21 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
8618, 85nfcxfr 2905 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
87 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑟
8884, 86, 87nfov 6872 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑎𝐹𝑟)
8988nfel1 2922 . . . . . . . . . . . . . . . . . 18 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
9083, 89nfim 1995 . . . . . . . . . . . . . . . . 17 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
91 sseq1 3786 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
92 xpeq12 5302 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9392anidms 562 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9493sseq2d 3793 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
95 weeq2 5266 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9691, 94, 953anbi123d 1560 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
97 breq2 4813 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
9896, 97anbi12d 624 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9915, 98syl5bb 274 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
10099anbi2d 622 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
101 oveq1 6849 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
102 difeq2 3884 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
103101, 102eleq12d 2838 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
104100, 103imbi12d 335 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
1055, 13, 14, 15, 16, 17, 18pwfseqlem3 9735 . . . . . . . . . . . . . . . . 17 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
10690, 104, 105chvar 2368 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
10774, 82, 106chvar 2368 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
108107eldifbd 3745 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎)
109108expr 448 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
11066, 109sylbird 251 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
111110con4d 115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω))
11248, 56, 111vtocl 3411 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω))
11347, 112vtoclg 3418 . . . . . . . . 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 8764 . . . . . . 7 (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
117115, 116sylibr 225 . . . . . 6 (𝜑𝑍 ∈ Fin)
1185, 13, 14, 15, 16, 17, 18pwfseqlem2 9734 . . . . . 6 ((𝑍 ∈ Fin ∧ (𝑊𝑍) ∈ V) → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
119117, 48, 118sylancl 580 . . . . 5 (𝜑 → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
120119, 23eqeltrrd 2845 . . . 4 (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
1214, 12, 24fpwwe2lem3 9708 . . . . . . . . . 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 678 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
123 cnvimass 5667 . . . . . . . . . . . 12 ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊𝑍)
12427simprd 489 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝑍) ⊆ (𝑍 × 𝑍))
125 dmss 5491 . . . . . . . . . . . . . 14 ((𝑊𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
126124, 125syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
127 dmxpss 5748 . . . . . . . . . . . . 13 dom (𝑍 × 𝑍) ⊆ 𝑍
128126, 127syl6ss 3773 . . . . . . . . . . . 12 (𝜑 → dom (𝑊𝑍) ⊆ 𝑍)
129123, 128syl5ss 3772 . . . . . . . . . . 11 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
130117, 129ssfid 8390 . . . . . . . . . 10 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
13148inex1 4960 . . . . . . . . . 10 ((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
1325, 13, 14, 15, 16, 17, 18pwfseqlem2 9734 . . . . . . . . . 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 580 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
134122, 133eqtr3d 2801 . . . . . . . 8 (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
135 f1of1 6319 . . . . . . . . . 10 (𝐻:ω–1-1-onto𝑋𝐻:ω–1-1𝑋)
13614, 135syl 17 . . . . . . . . 9 (𝜑𝐻:ω–1-1𝑋)
137 ficardom 9038 . . . . . . . . . 10 (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
138117, 137syl 17 . . . . . . . . 9 (𝜑 → (card‘𝑍) ∈ ω)
139 ficardom 9038 . . . . . . . . . 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 6711 . . . . . . . . 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 865 . . . . . . . 8 (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
143134, 142mpbid 223 . . . . . . 7 (𝜑 → (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))
144143eqcomd 2771 . . . . . 6 (𝜑 → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
145 finnum 9025 . . . . . . . 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 9025 . . . . . . . 8 (𝑍 ∈ Fin → 𝑍 ∈ dom card)
148117, 147syl 17 . . . . . . 7 (𝜑𝑍 ∈ dom card)
149 carden2 9064 . . . . . . 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 579 . . . . . 6 (𝜑 → ((card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
151144, 150mpbid 223 . . . . 5 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
152 dfpss2 3853 . . . . . . . 8 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
153152baib 531 . . . . . . 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 8353 . . . . . . . . 9 ((𝑍 ∈ Fin ∧ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
156 sdomnen 8189 . . . . . . . . 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 401 . . . . . . 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 251 . . . . 5 (𝜑 → (¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
161151, 160mt4d 153 . . . 4 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)
162120, 161eleqtrrd 2847 . . 3 (𝜑 → (𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))
163 fvex 6388 . . . 4 (𝐻‘(card‘𝑍)) ∈ V
164163eliniseg 5676 . . . 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 209 . 2 (𝜑 → (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
16726simprd 489 . . . . 5 (𝜑 → ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))
168167simpld 488 . . . 4 (𝜑 → (𝑊𝑍) We 𝑍)
169 weso 5268 . . . 4 ((𝑊𝑍) We 𝑍 → (𝑊𝑍) Or 𝑍)
170168, 169syl 17 . . 3 (𝜑 → (𝑊𝑍) Or 𝑍)
171 sonr 5219 . . 3 (((𝑊𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
172170, 120, 171syl2anc 579 . 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  [wsbc 3596  cdif 3729  cin 3731  wss 3732  wpss 3733  ifcif 4243  𝒫 cpw 4315  {csn 4334   cuni 4594   cint 4633   ciun 4676   class class class wbr 4809  {copab 4871   Or wor 5197   We wwe 5235   × cxp 5275  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  Oncon0 5908  1-1wf1 6065  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  ωcom 7263  𝑚 cmap 8060  cen 8157  cdom 8158  csdm 8159  Fincfn 8160  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-oi 8622  df-card 9016
This theorem is referenced by:  pwfseqlem5  9738
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