MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwfseqlem4a Structured version   Visualization version   GIF version

Theorem pwfseqlem4a 9771
Description: Lemma for pwfseqlem4 9772. (Contributed by Mario Carneiro, 7-Jun-2016.)
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})))
Assertion
Ref Expression
pwfseqlem4a ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑠,𝑎,𝐹   𝑤,𝐺   𝑤,𝐾   𝑟,𝑎,𝑥,𝑧,𝐻,𝑠   𝑛,𝑎,𝜑,𝑠,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑎,𝑛,𝑟,𝑠,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑠,𝑟,𝑎)   𝐴(𝑤)   𝐷(𝑥,𝑤,𝑠,𝑟,𝑎)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑛,𝑠,𝑟,𝑎)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑛,𝑠,𝑟,𝑎)   𝑋(𝑥,𝑧,𝑤,𝑛,𝑠,𝑟,𝑎)

Proof of Theorem pwfseqlem4a
StepHypRef Expression
1 isfinite 8799 . . 3 (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2 simpr 478 . . . . . . 7 ((𝜑𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3 vex 3388 . . . . . . 7 𝑠 ∈ V
4 pwfseqlem4.g . . . . . . . 8 (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
5 pwfseqlem4.x . . . . . . . 8 (𝜑𝑋𝐴)
6 pwfseqlem4.h . . . . . . . 8 (𝜑𝐻:ω–1-1-onto𝑋)
7 pwfseqlem4.ps . . . . . . . 8 (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
8 pwfseqlem4.k . . . . . . . 8 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
9 pwfseqlem4.d . . . . . . . 8 𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
10 pwfseqlem4.f . . . . . . . 8 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
114, 5, 6, 7, 8, 9, 10pwfseqlem2 9769 . . . . . . 7 ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
122, 3, 11sylancl 581 . . . . . 6 ((𝜑𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13 f1of 6356 . . . . . . . . 9 (𝐻:ω–1-1-onto𝑋𝐻:ω⟶𝑋)
146, 13syl 17 . . . . . . . 8 (𝜑𝐻:ω⟶𝑋)
1514, 5fssd 6270 . . . . . . 7 (𝜑𝐻:ω⟶𝐴)
16 ficardom 9073 . . . . . . 7 (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17 ffvelrn 6583 . . . . . . 7 ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
1815, 16, 17syl2an 590 . . . . . 6 ((𝜑𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
1912, 18eqeltrd 2878 . . . . 5 ((𝜑𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
2019ex 402 . . . 4 (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
2120adantr 473 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
221, 21syl5bir 235 . 2 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
23 omelon 8793 . . . . 5 ω ∈ On
24 onenon 9061 . . . . 5 (ω ∈ On → ω ∈ dom card)
2523, 24ax-mp 5 . . . 4 ω ∈ dom card
26 simpr3 1253 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
27 19.8a 2216 . . . . . 6 (𝑠 We 𝑎 → ∃𝑠 𝑠 We 𝑎)
2826, 27syl 17 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
29 ween 9144 . . . . 5 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
3028, 29sylibr 226 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
31 domtri2 9101 . . . 4 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
3225, 30, 31sylancr 582 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
33 nfv 2010 . . . . . . 7 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
34 nfcv 2941 . . . . . . . . 9 𝑟𝑎
35 nfmpt22 6957 . . . . . . . . . 10 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
3610, 35nfcxfr 2939 . . . . . . . . 9 𝑟𝐹
37 nfcv 2941 . . . . . . . . 9 𝑟𝑠
3834, 36, 37nfov 6908 . . . . . . . 8 𝑟(𝑎𝐹𝑠)
3938nfel1 2956 . . . . . . 7 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
4033, 39nfim 1996 . . . . . 6 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
41 sseq1 3822 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
42 weeq1 5300 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
4341, 423anbi23d 1564 . . . . . . . . 9 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
4443anbi1d 624 . . . . . . . 8 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
4544anbi2d 623 . . . . . . 7 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
46 oveq2 6886 . . . . . . . 8 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
4746eleq1d 2863 . . . . . . 7 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
4845, 47imbi12d 336 . . . . . 6 (𝑟 = 𝑠 → (((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))))
49 nfv 2010 . . . . . . . 8 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
50 nfcv 2941 . . . . . . . . . 10 𝑥𝑎
51 nfmpt21 6956 . . . . . . . . . . 11 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
5210, 51nfcxfr 2939 . . . . . . . . . 10 𝑥𝐹
53 nfcv 2941 . . . . . . . . . 10 𝑥𝑟
5450, 52, 53nfov 6908 . . . . . . . . 9 𝑥(𝑎𝐹𝑟)
5554nfel1 2956 . . . . . . . 8 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
5649, 55nfim 1996 . . . . . . 7 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
57 sseq1 3822 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
58 xpeq12 5337 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
5958anidms 563 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6059sseq2d 3829 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
61 weeq2 5301 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
6257, 60, 613anbi123d 1561 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
63 breq2 4847 . . . . . . . . . . 11 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
6462, 63anbi12d 625 . . . . . . . . . 10 (𝑥 = 𝑎 → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
657, 64syl5bb 275 . . . . . . . . 9 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
6665anbi2d 623 . . . . . . . 8 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
67 oveq1 6885 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
68 difeq2 3920 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
6967, 68eleq12d 2872 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
7066, 69imbi12d 336 . . . . . . 7 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
714, 5, 6, 7, 8, 9, 10pwfseqlem3 9770 . . . . . . 7 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
7256, 70, 71chvar 2402 . . . . . 6 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
7340, 48, 72chvar 2402 . . . . 5 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
7473eldifad 3781 . . . 4 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
7574expr 449 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → (𝑎𝐹𝑠) ∈ 𝐴))
7632, 75sylbird 252 . 2 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
7722, 76pm2.61d 172 1 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  {crab 3093  Vcvv 3385  cdif 3766  wss 3769  ifcif 4277  𝒫 cpw 4349   cint 4667   ciun 4710   class class class wbr 4843   We wwe 5270   × cxp 5310  ccnv 5311  dom cdm 5312  ran crn 5313  Oncon0 5941  wf 6097  1-1wf1 6098  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  cmpt2 6880  ωcom 7299  𝑚 cmap 8095  cdom 8193  csdm 8194  Fincfn 8195  cardccrd 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051
This theorem is referenced by:  pwfseqlem4  9772
  Copyright terms: Public domain W3C validator