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Theorem pwfseqlem2 9360
 Description: Lemma for pwfseq 9365. (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 6556 . . 3 (𝑎 = 𝑌 → (𝑎𝐹𝑠) = (𝑌𝐹𝑠))
2 fveq2 6103 . . . 4 (𝑎 = 𝑌 → (card‘𝑎) = (card‘𝑌))
32fveq2d 6107 . . 3 (𝑎 = 𝑌 → (𝐻‘(card‘𝑎)) = (𝐻‘(card‘𝑌)))
41, 3eqeq12d 2625 . 2 (𝑎 = 𝑌 → ((𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)) ↔ (𝑌𝐹𝑠) = (𝐻‘(card‘𝑌))))
5 oveq2 6557 . . 3 (𝑠 = 𝑅 → (𝑌𝐹𝑠) = (𝑌𝐹𝑅))
65eqeq1d 2612 . 2 (𝑠 = 𝑅 → ((𝑌𝐹𝑠) = (𝐻‘(card‘𝑌)) ↔ (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))))
7 nfcv 2751 . . 3 𝑥𝑎
8 nfcv 2751 . . 3 𝑟𝑎
9 nfcv 2751 . . 3 𝑟𝑠
10 pwfseqlem4.f . . . . . 6 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
11 nfmpt21 6620 . . . . . 6 𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
1210, 11nfcxfr 2749 . . . . 5 𝑥𝐹
13 nfcv 2751 . . . . 5 𝑥𝑟
147, 12, 13nfov 6575 . . . 4 𝑥(𝑎𝐹𝑟)
1514nfeq1 2764 . . 3 𝑥(𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))
16 nfmpt22 6621 . . . . . 6 𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
1710, 16nfcxfr 2749 . . . . 5 𝑟𝐹
188, 17, 9nfov 6575 . . . 4 𝑟(𝑎𝐹𝑠)
1918nfeq1 2764 . . 3 𝑟(𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))
20 oveq1 6556 . . . 4 (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
21 fveq2 6103 . . . . 5 (𝑥 = 𝑎 → (card‘𝑥) = (card‘𝑎))
2221fveq2d 6107 . . . 4 (𝑥 = 𝑎 → (𝐻‘(card‘𝑥)) = (𝐻‘(card‘𝑎)))
2320, 22eqeq12d 2625 . . 3 (𝑥 = 𝑎 → ((𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)) ↔ (𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))))
24 oveq2 6557 . . . 4 (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
2524eqeq1d 2612 . . 3 (𝑟 = 𝑠 → ((𝑎𝐹𝑟) = (𝐻‘(card‘𝑎)) ↔ (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))))
26 vex 3176 . . . . . 6 𝑥 ∈ V
27 vex 3176 . . . . . 6 𝑟 ∈ V
28 fvex 6113 . . . . . . 7 (𝐻‘(card‘𝑥)) ∈ V
29 fvex 6113 . . . . . . 7 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
3028, 29ifex 4106 . . . . . 6 if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V
3110ovmpt4g 6681 . . . . . 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 1416 . . . . 5 (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
33 iftrue 4042 . . . . 5 (𝑥 ∈ Fin → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐻‘(card‘𝑥)))
3432, 33syl5eq 2656 . . . 4 (𝑥 ∈ Fin → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
3534adantr 480 . . 3 ((𝑥 ∈ Fin ∧ 𝑟𝑉) → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
367, 8, 9, 15, 19, 23, 25, 35vtocl2gaf 3246 . 2 ((𝑎 ∈ Fin ∧ 𝑠𝑉) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
374, 6, 36vtocl2ga 3247 1 ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108  ∩ cint 4410  ∪ ciun 4455   class class class wbr 4583   We wwe 4996   × cxp 5036  ◡ccnv 5037  ran crn 5039  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ↑𝑚 cmap 7744   ≼ cdom 7839  Fincfn 7841  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  pwfseqlem4a  9362  pwfseqlem4  9363
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