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Theorem pwfseqlem4a 9480
Description: Lemma for pwfseqlem4 9481. (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 8546 . . 3 (𝑎 ∈ Fin ↔ 𝑎 ≺ ω)
2 simpr 477 . . . . . . 7 ((𝜑𝑎 ∈ Fin) → 𝑎 ∈ Fin)
3 vex 3201 . . . . . . 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 9478 . . . . . . 7 ((𝑎 ∈ Fin ∧ 𝑠 ∈ V) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
122, 3, 11sylancl 694 . . . . . 6 ((𝜑𝑎 ∈ Fin) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
13 f1of 6135 . . . . . . . . 9 (𝐻:ω–1-1-onto𝑋𝐻:ω⟶𝑋)
146, 13syl 17 . . . . . . . 8 (𝜑𝐻:ω⟶𝑋)
1514, 5fssd 6055 . . . . . . 7 (𝜑𝐻:ω⟶𝐴)
16 ficardom 8784 . . . . . . 7 (𝑎 ∈ Fin → (card‘𝑎) ∈ ω)
17 ffvelrn 6355 . . . . . . 7 ((𝐻:ω⟶𝐴 ∧ (card‘𝑎) ∈ ω) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
1815, 16, 17syl2an 494 . . . . . 6 ((𝜑𝑎 ∈ Fin) → (𝐻‘(card‘𝑎)) ∈ 𝐴)
1912, 18eqeltrd 2700 . . . . 5 ((𝜑𝑎 ∈ Fin) → (𝑎𝐹𝑠) ∈ 𝐴)
2019ex 450 . . . 4 (𝜑 → (𝑎 ∈ Fin → (𝑎𝐹𝑠) ∈ 𝐴))
2120adantr 481 . . 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 8540 . . . . 5 ω ∈ On
24 onenon 8772 . . . . 5 (ω ∈ On → ω ∈ dom card)
2523, 24ax-mp 5 . . . 4 ω ∈ dom card
26 simpr3 1068 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
27 19.8a 2051 . . . . . 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 8855 . . . . 5 (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
3028, 29sylibr 224 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
31 domtri2 8812 . . . 4 ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
3225, 30, 31sylancr 695 . . 3 ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
33 nfv 1842 . . . . . . 7 𝑟(𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
34 nfcv 2763 . . . . . . . . 9 𝑟𝑎
35 nfmpt22 6720 . . . . . . . . . 10 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
3610, 35nfcxfr 2761 . . . . . . . . 9 𝑟𝐹
37 nfcv 2763 . . . . . . . . 9 𝑟𝑠
3834, 36, 37nfov 6673 . . . . . . . 8 𝑟(𝑎𝐹𝑠)
3938nfel1 2778 . . . . . . 7 𝑟(𝑎𝐹𝑠) ∈ (𝐴𝑎)
4033, 39nfim 1824 . . . . . 6 𝑟((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
41 sseq1 3624 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
42 weeq1 5100 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
4341, 423anbi23d 1401 . . . . . . . . 9 (𝑟 = 𝑠 → ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
4443anbi1d 741 . . . . . . . 8 (𝑟 = 𝑠 → (((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
4544anbi2d 740 . . . . . . 7 (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
46 oveq2 6655 . . . . . . . 8 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
4746eleq1d 2685 . . . . . . 7 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴𝑎)))
4845, 47imbi12d 334 . . . . . 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 1842 . . . . . . . 8 𝑥(𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
50 nfcv 2763 . . . . . . . . . 10 𝑥𝑎
51 nfmpt21 6719 . . . . . . . . . . 11 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
5210, 51nfcxfr 2761 . . . . . . . . . 10 𝑥𝐹
53 nfcv 2763 . . . . . . . . . 10 𝑥𝑟
5450, 52, 53nfov 6673 . . . . . . . . 9 𝑥(𝑎𝐹𝑟)
5554nfel1 2778 . . . . . . . 8 𝑥(𝑎𝐹𝑟) ∈ (𝐴𝑎)
5649, 55nfim 1824 . . . . . . 7 𝑥((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
57 sseq1 3624 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
58 xpeq12 5132 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
5958anidms 677 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6059sseq2d 3631 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
61 weeq2 5101 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
6257, 60, 613anbi123d 1398 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
63 breq2 4655 . . . . . . . . . . 11 (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
6462, 63anbi12d 747 . . . . . . . . . 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 740 . . . . . . . 8 (𝑥 = 𝑎 → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
67 oveq1 6654 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
68 difeq2 3720 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
6967, 68eleq12d 2694 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴𝑎)))
7066, 69imbi12d 334 . . . . . . 7 (𝑥 = 𝑎 → (((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥)) ↔ ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))))
714, 5, 6, 7, 8, 9, 10pwfseqlem3 9479 . . . . . . 7 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
7256, 70, 71chvar 2261 . . . . . 6 ((𝜑 ∧ ((𝑎𝐴𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴𝑎))
7340, 48, 72chvar 2261 . . . . 5 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴𝑎))
7473eldifad 3584 . . . 4 ((𝜑 ∧ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
7574expr 643 . . 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 170 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 384  w3a 1037   = wceq 1482  wex 1703  wcel 1989  {crab 2915  Vcvv 3198  cdif 3569  wss 3572  ifcif 4084  𝒫 cpw 4156   cint 4473   ciun 4518   class class class wbr 4651   We wwe 5070   × cxp 5110  ccnv 5111  dom cdm 5112  ran crn 5113  Oncon0 5721  wf 5882  1-1wf1 5883  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  cmpt2 6649  ωcom 7062  𝑚 cmap 7854  cdom 7950  csdm 7951  Fincfn 7952  cardccrd 8758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762
This theorem is referenced by:  pwfseqlem4  9481
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