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Theorem pwfseqlem2 9441
Description: Lemma for pwfseq 9446. (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 6622 . . 3 (𝑎 = 𝑌 → (𝑎𝐹𝑠) = (𝑌𝐹𝑠))
2 fveq2 6158 . . . 4 (𝑎 = 𝑌 → (card‘𝑎) = (card‘𝑌))
32fveq2d 6162 . . 3 (𝑎 = 𝑌 → (𝐻‘(card‘𝑎)) = (𝐻‘(card‘𝑌)))
41, 3eqeq12d 2636 . 2 (𝑎 = 𝑌 → ((𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)) ↔ (𝑌𝐹𝑠) = (𝐻‘(card‘𝑌))))
5 oveq2 6623 . . 3 (𝑠 = 𝑅 → (𝑌𝐹𝑠) = (𝑌𝐹𝑅))
65eqeq1d 2623 . 2 (𝑠 = 𝑅 → ((𝑌𝐹𝑠) = (𝐻‘(card‘𝑌)) ↔ (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))))
7 nfcv 2761 . . 3 𝑥𝑎
8 nfcv 2761 . . 3 𝑟𝑎
9 nfcv 2761 . . 3 𝑟𝑠
10 pwfseqlem4.f . . . . . 6 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
11 nfmpt21 6687 . . . . . 6 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
1210, 11nfcxfr 2759 . . . . 5 𝑥𝐹
13 nfcv 2761 . . . . 5 𝑥𝑟
147, 12, 13nfov 6641 . . . 4 𝑥(𝑎𝐹𝑟)
1514nfeq1 2774 . . 3 𝑥(𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))
16 nfmpt22 6688 . . . . . 6 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
1710, 16nfcxfr 2759 . . . . 5 𝑟𝐹
188, 17, 9nfov 6641 . . . 4 𝑟(𝑎𝐹𝑠)
1918nfeq1 2774 . . 3 𝑟(𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))
20 oveq1 6622 . . . 4 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
21 fveq2 6158 . . . . 5 (𝑥 = 𝑎 → (card‘𝑥) = (card‘𝑎))
2221fveq2d 6162 . . . 4 (𝑥 = 𝑎 → (𝐻‘(card‘𝑥)) = (𝐻‘(card‘𝑎)))
2320, 22eqeq12d 2636 . . 3 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)) ↔ (𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))))
24 oveq2 6623 . . . 4 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
2524eqeq1d 2623 . . 3 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) = (𝐻‘(card‘𝑎)) ↔ (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))))
26 vex 3193 . . . . . 6 𝑥 ∈ V
27 vex 3193 . . . . . 6 𝑟 ∈ V
28 fvex 6168 . . . . . . 7 (𝐻‘(card‘𝑥)) ∈ V
29 fvex 6168 . . . . . . 7 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
3028, 29ifex 4134 . . . . . 6 if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V
3110ovmpt4g 6748 . . . . . 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})))
3226, 27, 30, 31mp3an 1421 . . . . 5 (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
33 iftrue 4070 . . . . 5 (𝑥 ∈ Fin → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐻‘(card‘𝑥)))
3432, 33syl5eq 2667 . . . 4 (𝑥 ∈ Fin → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
3534adantr 481 . . 3 ((𝑥 ∈ Fin ∧ 𝑟𝑉) → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
367, 8, 9, 15, 19, 23, 25, 35vtocl2gaf 3263 . 2 ((𝑎 ∈ Fin ∧ 𝑠𝑉) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
374, 6, 36vtocl2ga 3264 1 ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  wss 3560  ifcif 4064  𝒫 cpw 4136   cint 4447   ciun 4492   class class class wbr 4623   We wwe 5042   × cxp 5082  ccnv 5083  ran crn 5085  1-1wf1 5854  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  cmpt2 6617  ωcom 7027  𝑚 cmap 7817  cdom 7913  Fincfn 7915  cardccrd 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  pwfseqlem4a  9443  pwfseqlem4  9444
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