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Theorem pwfseqlem4 9686
Description: Lemma for pwfseq 9688. Derive a final contradiction from the function 𝐹 in pwfseqlem3 9684. Applying fpwwe2 9667 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 2771 . . . . . . . . . . 11 𝑍 = 𝑍
2 eqid 2771 . . . . . . . . . . 11 (𝑊𝑍) = (𝑊𝑍)
31, 2pm3.2i 447 . . . . . . . . . 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 8704 . . . . . . . . . . . . . 14 ω ∈ V
7 ovex 6823 . . . . . . . . . . . . . 14 (𝐴𝑚 𝑛) ∈ V
86, 7iunex 7294 . . . . . . . . . . . . 13 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
9 f1dmex 7283 . . . . . . . . . . . . 13 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
105, 8, 9sylancl 574 . . . . . . . . . . . 12 (𝜑 → 𝒫 𝐴 ∈ V)
11 pwexb 7122 . . . . . . . . . . . 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 9685 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20 pwfseqlem4.z . . . . . . . . . . 11 𝑍 = dom 𝑊
214, 12, 19, 20fpwwe2 9667 . . . . . . . . . 10 (𝜑 → ((𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊𝑍) = (𝑊𝑍))))
223, 21mpbiri 248 . . . . . . . . 9 (𝜑 → (𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
2322simprd 483 . . . . . . . 8 (𝜑 → (𝑍𝐹(𝑊𝑍)) ∈ 𝑍)
2422simpld 482 . . . . . . . . . . . . 13 (𝜑𝑍𝑊(𝑊𝑍))
254, 12fpwwe2lem2 9656 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 (𝜑 → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
2827simpld 482 . . . . . . . . . 10 (𝜑𝑍𝐴)
2912, 28ssexd 4939 . . . . . . . . 9 (𝜑𝑍 ∈ V)
30 sseq1 3775 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → (𝑎𝐴𝑍𝐴))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑍𝑎 = 𝑍)
3231sqxpeqd 5281 . . . . . . . . . . . . . . 15 (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
3332sseq2d 3782 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
34 weeq2 5238 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) We 𝑎 ↔ (𝑊𝑍) We 𝑍))
3530, 33, 343anbi123d 1547 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → ((𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎) ↔ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
3635anbi2d 614 . . . . . . . . . . . 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 1111 . . . . . . . . . . . . . . 15 (((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊𝑍) We 𝑍) → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
3938adantrr 696 . . . . . . . . . . . . . 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 549 . . . . . . . . . . . 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 6800 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → (𝑎𝐹(𝑊𝑍)) = (𝑍𝐹(𝑊𝑍)))
4443, 31eleq12d 2844 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
45 breq1 4789 . . . . . . . . . . . 12 (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
4644, 45imbi12d 333 . . . . . . . . . . 11 (𝑎 = 𝑍 → (((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω)))
4742, 46imbi12d 333 . . . . . . . . . 10 (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω))))
48 fvex 6342 . . . . . . . . . . 11 (𝑊𝑍) ∈ V
49 sseq1 3775 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑎 × 𝑎)))
50 weeq1 5237 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 We 𝑎 ↔ (𝑊𝑍) We 𝑎))
5149, 503anbi23d 1550 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)))
5251anbi2d 614 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎))))
53 oveq2 6801 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊𝑍)))
5453eleq1d 2835 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊𝑍)) ∈ 𝑎))
5554imbi1d 330 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)))
5652, 55imbi12d 333 . . . . . . . . . . 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 8707 . . . . . . . . . . . . . . 15 ω ∈ On
58 onenon 8975 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
5957, 58ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
60 simpr3 1237 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
61 19.8a 2206 . . . . . . . . . . . . . . . 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 9058 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
6462, 63sylibr 224 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
65 domtri2 9015 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
6659, 64, 65sylancr 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
67 nfv 1995 . . . . . . . . . . . . . . . . 17 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
68 nfcv 2913 . . . . . . . . . . . . . . . . . . 19 𝑟𝑎
69 nfmpt22 6870 . . . . . . . . . . . . . . . . . . . 20 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
7018, 69nfcxfr 2911 . . . . . . . . . . . . . . . . . . 19 𝑟𝐹
71 nfcv 2913 . . . . . . . . . . . . . . . . . . 19 𝑟𝑠
7268, 70, 71nfov 6821 . . . . . . . . . . . . . . . . . 18 𝑟(𝑎𝐹𝑠)
7372nfel1 2928 . . . . . . . . . . . . . . . . 17 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
7467, 73nfim 1977 . . . . . . . . . . . . . . . 16 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
75 sseq1 3775 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
76 weeq1 5237 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
7775, 763anbi23d 1550 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
7877anbi1d 615 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
7978anbi2d 614 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
80 oveq2 6801 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
8180eleq1d 2835 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
8279, 81imbi12d 333 . . . . . . . . . . . . . . . 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 1995 . . . . . . . . . . . . . . . . . 18 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
84 nfcv 2913 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎
85 nfmpt21 6869 . . . . . . . . . . . . . . . . . . . . 21 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
8618, 85nfcxfr 2911 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
87 nfcv 2913 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑟
8884, 86, 87nfov 6821 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑎𝐹𝑟)
8988nfel1 2928 . . . . . . . . . . . . . . . . . 18 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
9083, 89nfim 1977 . . . . . . . . . . . . . . . . 17 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
91 sseq1 3775 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
92 xpeq12 5274 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9392anidms 556 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9493sseq2d 3782 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
95 weeq2 5238 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9691, 94, 953anbi123d 1547 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
97 breq2 4790 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
9896, 97anbi12d 616 . . . . . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
101 oveq1 6800 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
102 difeq2 3873 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
103101, 102eleq12d 2844 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
104100, 103imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
1055, 13, 14, 15, 16, 17, 18pwfseqlem3 9684 . . . . . . . . . . . . . . . . 17 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
10690, 104, 105chvar 2424 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
10774, 82, 106chvar 2424 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
108107eldifbd 3736 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎)
109108expr 444 . . . . . . . . . . . . 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 115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω))
11248, 56, 111vtocl 3410 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω))
11347, 112vtoclg 3417 . . . . . . . . 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 8713 . . . . . . 7 (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
117115, 116sylibr 224 . . . . . 6 (𝜑𝑍 ∈ Fin)
1185, 13, 14, 15, 16, 17, 18pwfseqlem2 9683 . . . . . 6 ((𝑍 ∈ Fin ∧ (𝑊𝑍) ∈ V) → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
119117, 48, 118sylancl 574 . . . . 5 (𝜑 → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
120119, 23eqeltrrd 2851 . . . 4 (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
1214, 12, 24fpwwe2lem3 9657 . . . . . . . . . 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 667 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
123 cnvimass 5626 . . . . . . . . . . . 12 ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊𝑍)
12427simprd 483 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝑍) ⊆ (𝑍 × 𝑍))
125 dmss 5461 . . . . . . . . . . . . . 14 ((𝑊𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
126124, 125syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
127 dmxpss 5706 . . . . . . . . . . . . 13 dom (𝑍 × 𝑍) ⊆ 𝑍
128126, 127syl6ss 3764 . . . . . . . . . . . 12 (𝜑 → dom (𝑊𝑍) ⊆ 𝑍)
129123, 128syl5ss 3763 . . . . . . . . . . 11 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
130117, 129ssfid 8339 . . . . . . . . . 10 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
13148inex1 4933 . . . . . . . . . 10 ((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
1325, 13, 14, 15, 16, 17, 18pwfseqlem2 9683 . . . . . . . . . 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 574 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
134122, 133eqtr3d 2807 . . . . . . . 8 (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
135 f1of1 6277 . . . . . . . . . 10 (𝐻:ω–1-1-onto𝑋𝐻:ω–1-1𝑋)
13614, 135syl 17 . . . . . . . . 9 (𝜑𝐻:ω–1-1𝑋)
137 ficardom 8987 . . . . . . . . . 10 (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
138117, 137syl 17 . . . . . . . . 9 (𝜑 → (card‘𝑍) ∈ ω)
139 ficardom 8987 . . . . . . . . . 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 6662 . . . . . . . . 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 1474 . . . . . . . 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 2777 . . . . . 6 (𝜑 → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
145 finnum 8974 . . . . . . . 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 8974 . . . . . . . 8 (𝑍 ∈ Fin → 𝑍 ∈ dom card)
148117, 147syl 17 . . . . . . 7 (𝜑𝑍 ∈ dom card)
149 carden2 9013 . . . . . . 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 573 . . . . . 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 3842 . . . . . . . 8 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
153152baib 525 . . . . . . 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 8302 . . . . . . . . 9 ((𝑍 ∈ Fin ∧ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
156 sdomnen 8138 . . . . . . . . 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 397 . . . . . . 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 153 . . . 4 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)
162120, 161eleqtrrd 2853 . . 3 (𝜑 → (𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))
163 fvex 6342 . . . 4 (𝐻‘(card‘𝑍)) ∈ V
164163eliniseg 5635 . . . 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 483 . . . . 5 (𝜑 → ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))
168167simpld 482 . . . 4 (𝜑 → (𝑊𝑍) We 𝑍)
169 weso 5240 . . . 4 ((𝑊𝑍) We 𝑍 → (𝑊𝑍) Or 𝑍)
170168, 169syl 17 . . 3 (𝜑 → (𝑊𝑍) Or 𝑍)
171 sonr 5191 . . 3 (((𝑊𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
172170, 120, 171syl2anc 573 . 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  [wsbc 3587  cdif 3720  cin 3722  wss 3723  wpss 3724  ifcif 4225  𝒫 cpw 4297  {csn 4316   cuni 4574   cint 4611   ciun 4654   class class class wbr 4786  {copab 4846   Or wor 5169   We wwe 5207   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cima 5252  Oncon0 5866  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  ωcom 7212  𝑚 cmap 8009  cen 8106  cdom 8107  csdm 8108  Fincfn 8109  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-oi 8571  df-card 8965
This theorem is referenced by:  pwfseqlem5  9687
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