MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwfseqlem4 Structured version   Visualization version   GIF version

Theorem pwfseqlem4 10077
Description: Lemma for pwfseq 10079. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10075. Applying fpwwe2 10058 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 2801 . . . . . . . . . . 11 𝑍 = 𝑍
2 eqid 2801 . . . . . . . . . . 11 (𝑊𝑍) = (𝑊𝑍)
31, 2pm3.2i 474 . . . . . . . . . 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 9094 . . . . . . . . . . . . . 14 ω ∈ V
7 ovex 7172 . . . . . . . . . . . . . 14 (𝐴m 𝑛) ∈ V
86, 7iunex 7655 . . . . . . . . . . . . 13 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
9 f1dmex 7644 . . . . . . . . . . . . 13 ((𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑛 ∈ ω (𝐴m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
105, 8, 9sylancl 589 . . . . . . . . . . . 12 (𝜑 → 𝒫 𝐴 ∈ V)
11 pwexb 7472 . . . . . . . . . . . 12 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
1210, 11sylibr 237 . . . . . . . . . . 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 10076 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20 pwfseqlem4.z . . . . . . . . . . 11 𝑍 = dom 𝑊
214, 12, 19, 20fpwwe2 10058 . . . . . . . . . 10 (𝜑 → ((𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊𝑍) = (𝑊𝑍))))
223, 21mpbiri 261 . . . . . . . . 9 (𝜑 → (𝑍𝑊(𝑊𝑍) ∧ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
2322simprd 499 . . . . . . . 8 (𝜑 → (𝑍𝐹(𝑊𝑍)) ∈ 𝑍)
2422simpld 498 . . . . . . . . . . . . 13 (𝜑𝑍𝑊(𝑊𝑍))
254, 12fpwwe2lem2 10047 . . . . . . . . . . . . 13 (𝜑 → (𝑍𝑊(𝑊𝑍) ↔ ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))))
2624, 25mpbid 235 . . . . . . . . . . . 12 (𝜑 → ((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)))
2726simpld 498 . . . . . . . . . . 11 (𝜑 → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
2827simpld 498 . . . . . . . . . 10 (𝜑𝑍𝐴)
2912, 28ssexd 5195 . . . . . . . . 9 (𝜑𝑍 ∈ V)
30 sseq1 3943 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → (𝑎𝐴𝑍𝐴))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑍𝑎 = 𝑍)
3231sqxpeqd 5555 . . . . . . . . . . . . . . 15 (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
3332sseq2d 3950 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑍 × 𝑍)))
34 weeq2 5512 . . . . . . . . . . . . . 14 (𝑎 = 𝑍 → ((𝑊𝑍) We 𝑎 ↔ (𝑊𝑍) We 𝑍))
3530, 33, 343anbi123d 1433 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → ((𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎) ↔ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
3635anbi2d 631 . . . . . . . . . . . 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 1115 . . . . . . . . . . . . . . 15 (((𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊𝑍) We 𝑍) → (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍))
3938adantrr 716 . . . . . . . . . . . . . 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 563 . . . . . . . . . . . 12 (𝜑 ↔ (𝜑 ∧ (𝑍𝐴 ∧ (𝑊𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊𝑍) We 𝑍)))
4236, 41syl6bbr 292 . . . . . . . . . . 11 (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) ↔ 𝜑))
43 oveq1 7146 . . . . . . . . . . . . 13 (𝑎 = 𝑍 → (𝑎𝐹(𝑊𝑍)) = (𝑍𝐹(𝑊𝑍)))
4443, 31eleq12d 2887 . . . . . . . . . . . 12 (𝑎 = 𝑍 → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊𝑍)) ∈ 𝑍))
45 breq1 5036 . . . . . . . . . . . 12 (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
4644, 45imbi12d 348 . . . . . . . . . . 11 (𝑎 = 𝑍 → (((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω)))
4742, 46imbi12d 348 . . . . . . . . . 10 (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω))))
48 fvex 6662 . . . . . . . . . . 11 (𝑊𝑍) ∈ V
49 sseq1 3943 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊𝑍) ⊆ (𝑎 × 𝑎)))
50 weeq1 5511 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑠 We 𝑎 ↔ (𝑊𝑍) We 𝑎))
5149, 503anbi23d 1436 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)))
5251anbi2d 631 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎))))
53 oveq2 7147 . . . . . . . . . . . . . 14 (𝑠 = (𝑊𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊𝑍)))
5453eleq1d 2877 . . . . . . . . . . . . 13 (𝑠 = (𝑊𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊𝑍)) ∈ 𝑎))
5554imbi1d 345 . . . . . . . . . . . 12 (𝑠 = (𝑊𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω)))
5652, 55imbi12d 348 . . . . . . . . . . 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 9097 . . . . . . . . . . . . . . 15 ω ∈ On
58 onenon 9366 . . . . . . . . . . . . . . 15 (ω ∈ On → ω ∈ dom card)
5957, 58ax-mp 5 . . . . . . . . . . . . . 14 ω ∈ dom card
60 simpr3 1193 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
616019.8ad 2180 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
62 ween 9450 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
6361, 62sylibr 237 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
64 domtri2 9406 . . . . . . . . . . . . . 14 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
6559, 63, 64sylancr 590 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
66 nfv 1915 . . . . . . . . . . . . . . . . 17 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
67 nfcv 2958 . . . . . . . . . . . . . . . . . . 19 𝑟𝑎
68 nfmpo2 7218 . . . . . . . . . . . . . . . . . . . 20 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
6918, 68nfcxfr 2956 . . . . . . . . . . . . . . . . . . 19 𝑟𝐹
70 nfcv 2958 . . . . . . . . . . . . . . . . . . 19 𝑟𝑠
7167, 69, 70nfov 7169 . . . . . . . . . . . . . . . . . 18 𝑟(𝑎𝐹𝑠)
7271nfel1 2974 . . . . . . . . . . . . . . . . 17 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
7366, 72nfim 1897 . . . . . . . . . . . . . . . 16 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
74 sseq1 3943 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
75 weeq1 5511 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
7674, 753anbi23d 1436 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
7776anbi1d 632 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
7877anbi2d 631 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
79 oveq2 7147 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
8079eleq1d 2877 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
8178, 80imbi12d 348 . . . . . . . . . . . . . . . 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))))
82 nfv 1915 . . . . . . . . . . . . . . . . . 18 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
83 nfcv 2958 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎
84 nfmpo1 7217 . . . . . . . . . . . . . . . . . . . . 21 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
8518, 84nfcxfr 2956 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
86 nfcv 2958 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑟
8783, 85, 86nfov 7169 . . . . . . . . . . . . . . . . . . 19 𝑥(𝑎𝐹𝑟)
8887nfel1 2974 . . . . . . . . . . . . . . . . . 18 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
8982, 88nfim 1897 . . . . . . . . . . . . . . . . 17 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
90 sseq1 3943 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
91 xpeq12 5548 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9291anidms 570 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
9392sseq2d 3950 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
94 weeq2 5512 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9590, 93, 943anbi123d 1433 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
96 breq2 5037 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
9795, 96anbi12d 633 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9815, 97syl5bb 286 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
9998anbi2d 631 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
100 oveq1 7146 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
101 difeq2 4047 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
102100, 101eleq12d 2887 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
10399, 102imbi12d 348 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
1045, 13, 14, 15, 16, 17, 18pwfseqlem3 10075 . . . . . . . . . . . . . . . . 17 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
10589, 103, 104chvarfv 2241 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
10673, 81, 105chvarfv 2241 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
107106eldifbd 3897 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎)
108107expr 460 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
10965, 108sylbird 263 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
110109con4d 115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎𝑎 ≺ ω))
11148, 56, 110vtocl 3510 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴 ∧ (𝑊𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊𝑍) We 𝑎)) → ((𝑎𝐹(𝑊𝑍)) ∈ 𝑎𝑎 ≺ ω))
11247, 111vtoclg 3518 . . . . . . . . 9 (𝑍 ∈ V → (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω)))
11329, 112mpcom 38 . . . . . . . 8 (𝜑 → ((𝑍𝐹(𝑊𝑍)) ∈ 𝑍𝑍 ≺ ω))
11423, 113mpd 15 . . . . . . 7 (𝜑𝑍 ≺ ω)
115 isfinite 9103 . . . . . . 7 (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
116114, 115sylibr 237 . . . . . 6 (𝜑𝑍 ∈ Fin)
1175, 13, 14, 15, 16, 17, 18pwfseqlem2 10074 . . . . . 6 ((𝑍 ∈ Fin ∧ (𝑊𝑍) ∈ V) → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
118116, 48, 117sylancl 589 . . . . 5 (𝜑 → (𝑍𝐹(𝑊𝑍)) = (𝐻‘(card‘𝑍)))
119118, 23eqeltrrd 2894 . . . 4 (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
1204, 12, 24fpwwe2lem3 10048 . . . . . . . . . 10 ((𝜑 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
121119, 120mpdan 686 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
122 cnvimass 5920 . . . . . . . . . . . 12 ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊𝑍)
12327simprd 499 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝑍) ⊆ (𝑍 × 𝑍))
124 dmss 5739 . . . . . . . . . . . . . 14 ((𝑊𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
125123, 124syl 17 . . . . . . . . . . . . 13 (𝜑 → dom (𝑊𝑍) ⊆ dom (𝑍 × 𝑍))
126 dmxpss 5999 . . . . . . . . . . . . 13 dom (𝑍 × 𝑍) ⊆ 𝑍
127125, 126sstrdi 3930 . . . . . . . . . . . 12 (𝜑 → dom (𝑊𝑍) ⊆ 𝑍)
128122, 127sstrid 3929 . . . . . . . . . . 11 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
129116, 128ssfid 8729 . . . . . . . . . 10 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
13048inex1 5188 . . . . . . . . . 10 ((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
1315, 13, 14, 15, 16, 17, 18pwfseqlem2 10074 . . . . . . . . . 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))}))))
132129, 130, 131sylancl 589 . . . . . . . . 9 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊𝑍) ∩ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) × ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
133121, 132eqtr3d 2838 . . . . . . . 8 (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
134 f1of1 6593 . . . . . . . . . 10 (𝐻:ω–1-1-onto𝑋𝐻:ω–1-1𝑋)
13514, 134syl 17 . . . . . . . . 9 (𝜑𝐻:ω–1-1𝑋)
136 ficardom 9378 . . . . . . . . . 10 (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
137116, 136syl 17 . . . . . . . . 9 (𝜑 → (card‘𝑍) ∈ ω)
138 ficardom 9378 . . . . . . . . . 10 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
139129, 138syl 17 . . . . . . . . 9 (𝜑 → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
140 f1fveq 7002 . . . . . . . . 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))}))))
141135, 137, 139, 140syl12anc 835 . . . . . . . 8 (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))))
142133, 141mpbid 235 . . . . . . 7 (𝜑 → (card‘𝑍) = (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})))
143142eqcomd 2807 . . . . . 6 (𝜑 → (card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
144 finnum 9365 . . . . . . . 8 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
145129, 144syl 17 . . . . . . 7 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
146 finnum 9365 . . . . . . . 8 (𝑍 ∈ Fin → 𝑍 ∈ dom card)
147116, 146syl 17 . . . . . . 7 (𝜑𝑍 ∈ dom card)
148 carden2 9404 . . . . . . 7 ((((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card ∧ 𝑍 ∈ dom card) → ((card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
149145, 147, 148syl2anc 587 . . . . . 6 (𝜑 → ((card‘((𝑊𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
150143, 149mpbid 235 . . . . 5 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
151 dfpss2 4016 . . . . . . . 8 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
152151baib 539 . . . . . . 7 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
153128, 152syl 17 . . . . . 6 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
154 php3 8691 . . . . . . . . 9 ((𝑍 ∈ Fin ∧ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
155 sdomnen 8525 . . . . . . . . 9 (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
156154, 155syl 17 . . . . . . . 8 ((𝑍 ∈ Fin ∧ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
157156ex 416 . . . . . . 7 (𝑍 ∈ Fin → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
158116, 157syl 17 . . . . . 6 (𝜑 → (((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
159153, 158sylbird 263 . . . . 5 (𝜑 → (¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍 → ¬ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
160150, 159mt4d 117 . . . 4 (𝜑 → ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)
161119, 160eleqtrrd 2896 . . 3 (𝜑 → (𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}))
162 fvex 6662 . . . 4 (𝐻‘(card‘𝑍)) ∈ V
163162eliniseg 5929 . . . 4 ((𝐻‘(card‘𝑍)) ∈ V → ((𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍))))
164162, 163ax-mp 5 . . 3 ((𝐻‘(card‘𝑍)) ∈ ((𝑊𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
165161, 164sylib 221 . 2 (𝜑 → (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
16626simprd 499 . . . . 5 (𝜑 → ((𝑊𝑍) We 𝑍 ∧ ∀𝑏𝑍 [((𝑊𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))
167166simpld 498 . . . 4 (𝜑 → (𝑊𝑍) We 𝑍)
168 weso 5514 . . . 4 ((𝑊𝑍) We 𝑍 → (𝑊𝑍) Or 𝑍)
169167, 168syl 17 . . 3 (𝜑 → (𝑊𝑍) Or 𝑍)
170 sonr 5464 . . 3 (((𝑊𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
171169, 119, 170syl2anc 587 . 2 (𝜑 → ¬ (𝐻‘(card‘𝑍))(𝑊𝑍)(𝐻‘(card‘𝑍)))
172165, 171pm2.65i 197 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  wral 3109  {crab 3113  Vcvv 3444  [wsbc 3723  cdif 3881  cin 3883  wss 3884  wpss 3885  ifcif 4428  𝒫 cpw 4500  {csn 4528   cuni 4803   cint 4841   ciun 4884   class class class wbr 5033  {copab 5095   Or wor 5441   We wwe 5481   × cxp 5521  ccnv 5522  dom cdm 5523  ran crn 5524  cima 5526  Oncon0 6163  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cmpo 7141  ωcom 7564  m cmap 8393  cen 8493  cdom 8494  csdm 8495  Fincfn 8496  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-card 9356
This theorem is referenced by:  pwfseqlem5  10078
  Copyright terms: Public domain W3C validator