MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwfseqlem2 Structured version   Visualization version   GIF version

Theorem pwfseqlem2 10069
Description: Lemma for pwfseq 10074. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.)
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
pwfseqlem2 ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))
Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑤,𝐺   𝑤,𝐾   𝐻,𝑟,𝑥,𝑧   𝜑,𝑛,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑛,𝑟,𝑥,𝑧   𝑉,𝑟,𝑥
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑟)   𝐴(𝑤)   𝐷(𝑥,𝑤,𝑟)   𝑅(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑛,𝑟)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑛,𝑟)   𝑉(𝑧,𝑤,𝑛)   𝑋(𝑥,𝑧,𝑤,𝑛,𝑟)   𝑌(𝑥,𝑧,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem2
Dummy variables 𝑎 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7152 . . 3 (𝑎 = 𝑌 → (𝑎𝐹𝑠) = (𝑌𝐹𝑠))
2 2fveq3 6668 . . 3 (𝑎 = 𝑌 → (𝐻‘(card‘𝑎)) = (𝐻‘(card‘𝑌)))
31, 2eqeq12d 2834 . 2 (𝑎 = 𝑌 → ((𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)) ↔ (𝑌𝐹𝑠) = (𝐻‘(card‘𝑌))))
4 oveq2 7153 . . 3 (𝑠 = 𝑅 → (𝑌𝐹𝑠) = (𝑌𝐹𝑅))
54eqeq1d 2820 . 2 (𝑠 = 𝑅 → ((𝑌𝐹𝑠) = (𝐻‘(card‘𝑌)) ↔ (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))))
6 nfcv 2974 . . 3 𝑥𝑎
7 nfcv 2974 . . 3 𝑟𝑎
8 nfcv 2974 . . 3 𝑟𝑠
9 pwfseqlem4.f . . . . . 6 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
10 nfmpo1 7223 . . . . . 6 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
119, 10nfcxfr 2972 . . . . 5 𝑥𝐹
12 nfcv 2974 . . . . 5 𝑥𝑟
136, 11, 12nfov 7175 . . . 4 𝑥(𝑎𝐹𝑟)
1413nfeq1 2990 . . 3 𝑥(𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))
15 nfmpo2 7224 . . . . . 6 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
169, 15nfcxfr 2972 . . . . 5 𝑟𝐹
177, 16, 8nfov 7175 . . . 4 𝑟(𝑎𝐹𝑠)
1817nfeq1 2990 . . 3 𝑟(𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))
19 oveq1 7152 . . . 4 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
20 2fveq3 6668 . . . 4 (𝑥 = 𝑎 → (𝐻‘(card‘𝑥)) = (𝐻‘(card‘𝑎)))
2119, 20eqeq12d 2834 . . 3 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)) ↔ (𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))))
22 oveq2 7153 . . . 4 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
2322eqeq1d 2820 . . 3 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) = (𝐻‘(card‘𝑎)) ↔ (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))))
24 vex 3495 . . . . . 6 𝑥 ∈ V
25 vex 3495 . . . . . 6 𝑟 ∈ V
26 fvex 6676 . . . . . . 7 (𝐻‘(card‘𝑥)) ∈ V
27 fvex 6676 . . . . . . 7 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
2826, 27ifex 4511 . . . . . 6 if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V
299ovmpt4g 7286 . . . . . 6 ((𝑥 ∈ V ∧ 𝑟 ∈ V ∧ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V) → (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
3024, 25, 28, 29mp3an 1452 . . . . 5 (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
31 iftrue 4469 . . . . 5 (𝑥 ∈ Fin → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐻‘(card‘𝑥)))
3230, 31syl5eq 2865 . . . 4 (𝑥 ∈ Fin → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
3332adantr 481 . . 3 ((𝑥 ∈ Fin ∧ 𝑟𝑉) → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
346, 7, 8, 14, 18, 21, 23, 33vtocl2gaf 3573 . 2 ((𝑎 ∈ Fin ∧ 𝑠𝑉) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
353, 5, 34vtocl2ga 3572 1 ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  wss 3933  ifcif 4463  𝒫 cpw 4535   cint 4867   ciun 4910   class class class wbr 5057   We wwe 5506   × cxp 5546  ccnv 5547  ran crn 5549  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  ωcom 7569  m cmap 8395  cdom 8495  Fincfn 8497  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150
This theorem is referenced by:  pwfseqlem4a  10071  pwfseqlem4  10072
  Copyright terms: Public domain W3C validator