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Theorem pwfseqlem4a 9684
Description: Lemma for pwfseqlem4 9685. (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 8712 . . 3 (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2 simpr 471 . . . . . . 7 ((𝜑𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3 vex 3352 . . . . . . 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 9682 . . . . . . 7 ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
122, 3, 11sylancl 566 . . . . . 6 ((𝜑𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13 f1of 6278 . . . . . . . . 9 (𝐻:ω–1-1-onto𝑋𝐻:ω⟶𝑋)
146, 13syl 17 . . . . . . . 8 (𝜑𝐻:ω⟶𝑋)
1514, 5fssd 6197 . . . . . . 7 (𝜑𝐻:ω⟶𝐴)
16 ficardom 8986 . . . . . . 7 (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17 ffvelrn 6500 . . . . . . 7 ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
1815, 16, 17syl2an 575 . . . . . 6 ((𝜑𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
1912, 18eqeltrd 2849 . . . . 5 ((𝜑𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
2019ex 397 . . . 4 (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
2120adantr 466 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
221, 21syl5bir 233 . 2 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
23 omelon 8706 . . . . 5 ω ∈ On
24 onenon 8974 . . . . 5 (ω ∈ On → ω ∈ dom card)
2523, 24ax-mp 5 . . . 4 ω ∈ dom card
26 simpr3 1236 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
27 19.8a 2205 . . . . . 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 9057 . . . . 5 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
3028, 29sylibr 224 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
31 domtri2 9014 . . . 4 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
3225, 30, 31sylancr 567 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
33 nfv 1994 . . . . . . 7 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
34 nfcv 2912 . . . . . . . . 9 𝑟𝑎
35 nfmpt22 6869 . . . . . . . . . 10 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
3610, 35nfcxfr 2910 . . . . . . . . 9 𝑟𝐹
37 nfcv 2912 . . . . . . . . 9 𝑟𝑠
3834, 36, 37nfov 6820 . . . . . . . 8 𝑟(𝑎𝐹𝑠)
3938nfel1 2927 . . . . . . 7 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
4033, 39nfim 1976 . . . . . 6 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
41 sseq1 3773 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
42 weeq1 5237 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
4341, 423anbi23d 1549 . . . . . . . . 9 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
4443anbi1d 607 . . . . . . . 8 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
4544anbi2d 606 . . . . . . 7 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
46 oveq2 6800 . . . . . . . 8 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
4746eleq1d 2834 . . . . . . 7 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
4845, 47imbi12d 333 . . . . . 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 1994 . . . . . . . 8 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
50 nfcv 2912 . . . . . . . . . 10 𝑥𝑎
51 nfmpt21 6868 . . . . . . . . . . 11 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
5210, 51nfcxfr 2910 . . . . . . . . . 10 𝑥𝐹
53 nfcv 2912 . . . . . . . . . 10 𝑥𝑟
5450, 52, 53nfov 6820 . . . . . . . . 9 𝑥(𝑎𝐹𝑟)
5554nfel1 2927 . . . . . . . 8 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
5649, 55nfim 1976 . . . . . . 7 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
57 sseq1 3773 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
58 xpeq12 5274 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
5958anidms 548 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6059sseq2d 3780 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
61 weeq2 5238 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
6257, 60, 613anbi123d 1546 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
63 breq2 4788 . . . . . . . . . . 11 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
6462, 63anbi12d 608 . . . . . . . . . 10 (𝑥 = 𝑎 → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
657, 64syl5bb 272 . . . . . . . . 9 (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
6665anbi2d 606 . . . . . . . 8 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
67 oveq1 6799 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
68 difeq2 3871 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
6967, 68eleq12d 2843 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
7066, 69imbi12d 333 . . . . . . 7 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
714, 5, 6, 7, 8, 9, 10pwfseqlem3 9683 . . . . . . 7 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
7256, 70, 71chvar 2423 . . . . . 6 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
7340, 48, 72chvar 2423 . . . . 5 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
7473eldifad 3733 . . . 4 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
7574expr 444 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → (𝑎𝐹𝑠) ∈ 𝐴))
7632, 75sylbird 250 . 2 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → (𝑎𝐹𝑠) ∈ 𝐴))
7722, 76pm2.61d 171 1 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {crab 3064  Vcvv 3349  cdif 3718  wss 3721  ifcif 4223  𝒫 cpw 4295   cint 4609   ciun 4652   class class class wbr 4784   We wwe 5207   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  Oncon0 5866  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  cmpt2 6794  ωcom 7211  𝑚 cmap 8008  cdom 8106  csdm 8107  Fincfn 8108  cardccrd 8960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964
This theorem is referenced by:  pwfseqlem4  9685
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