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Theorem pwfseqlem2 10697
Description: Lemma for pwfseq 10702. (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 7438 . . 3 (𝑎 = 𝑌 → (𝑎𝐹𝑠) = (𝑌𝐹𝑠))
2 2fveq3 6912 . . 3 (𝑎 = 𝑌 → (𝐻‘(card‘𝑎)) = (𝐻‘(card‘𝑌)))
31, 2eqeq12d 2751 . 2 (𝑎 = 𝑌 → ((𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)) ↔ (𝑌𝐹𝑠) = (𝐻‘(card‘𝑌))))
4 oveq2 7439 . . 3 (𝑠 = 𝑅 → (𝑌𝐹𝑠) = (𝑌𝐹𝑅))
54eqeq1d 2737 . 2 (𝑠 = 𝑅 → ((𝑌𝐹𝑠) = (𝐻‘(card‘𝑌)) ↔ (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))))
6 nfcv 2903 . . 3 𝑥𝑎
7 nfcv 2903 . . 3 𝑟𝑎
8 nfcv 2903 . . 3 𝑟𝑠
9 pwfseqlem4.f . . . . . 6 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
10 nfmpo1 7513 . . . . . 6 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
119, 10nfcxfr 2901 . . . . 5 𝑥𝐹
12 nfcv 2903 . . . . 5 𝑥𝑟
136, 11, 12nfov 7461 . . . 4 𝑥(𝑎𝐹𝑟)
1413nfeq1 2919 . . 3 𝑥(𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))
15 nfmpo2 7514 . . . . . 6 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
169, 15nfcxfr 2901 . . . . 5 𝑟𝐹
177, 16, 8nfov 7461 . . . 4 𝑟(𝑎𝐹𝑠)
1817nfeq1 2919 . . 3 𝑟(𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))
19 oveq1 7438 . . . 4 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
20 2fveq3 6912 . . . 4 (𝑥 = 𝑎 → (𝐻‘(card‘𝑥)) = (𝐻‘(card‘𝑎)))
2119, 20eqeq12d 2751 . . 3 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)) ↔ (𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))))
22 oveq2 7439 . . . 4 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
2322eqeq1d 2737 . . 3 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) = (𝐻‘(card‘𝑎)) ↔ (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))))
24 vex 3482 . . . . . 6 𝑥 ∈ V
25 vex 3482 . . . . . 6 𝑟 ∈ V
26 fvex 6920 . . . . . . 7 (𝐻‘(card‘𝑥)) ∈ V
27 fvex 6920 . . . . . . 7 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
2826, 27ifex 4581 . . . . . 6 if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V
299ovmpt4g 7580 . . . . . 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 1460 . . . . 5 (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
31 iftrue 4537 . . . . 5 (𝑥 ∈ Fin → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐻‘(card‘𝑥)))
3230, 31eqtrid 2787 . . . 4 (𝑥 ∈ Fin → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
3332adantr 480 . . 3 ((𝑥 ∈ Fin ∧ 𝑟𝑉) → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
346, 7, 8, 14, 18, 21, 23, 33vtocl2gaf 3579 . 2 ((𝑎 ∈ Fin ∧ 𝑠𝑉) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
353, 5, 34vtocl2ga 3578 1 ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963  ifcif 4531  𝒫 cpw 4605   cint 4951   ciun 4996   class class class wbr 5148   We wwe 5640   × cxp 5687  ccnv 5688  ran crn 5690  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  m cmap 8865  cdom 8982  Fincfn 8984  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  pwfseqlem4a  10699  pwfseqlem4  10700
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