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Theorem pwfseqlem2 10656
Description: Lemma for pwfseq 10661. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.)
Hypotheses
Ref Expression
pwfseqlem4.g (πœ‘ β†’ 𝐺:𝒫 𝐴–1-1β†’βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
pwfseqlem4.x (πœ‘ β†’ 𝑋 βŠ† 𝐴)
pwfseqlem4.h (πœ‘ β†’ 𝐻:ω–1-1-onto→𝑋)
pwfseqlem4.ps (πœ“ ↔ ((π‘₯ βŠ† 𝐴 ∧ π‘Ÿ βŠ† (π‘₯ Γ— π‘₯) ∧ π‘Ÿ We π‘₯) ∧ Ο‰ β‰Ό π‘₯))
pwfseqlem4.k ((πœ‘ ∧ πœ“) β†’ 𝐾:βˆͺ 𝑛 ∈ Ο‰ (π‘₯ ↑m 𝑛)–1-1β†’π‘₯)
pwfseqlem4.d 𝐷 = (πΊβ€˜{𝑀 ∈ π‘₯ ∣ ((β—‘πΎβ€˜π‘€) ∈ ran 𝐺 ∧ Β¬ 𝑀 ∈ (β—‘πΊβ€˜(β—‘πΎβ€˜π‘€)))})
pwfseqlem4.f 𝐹 = (π‘₯ ∈ V, π‘Ÿ ∈ V ↦ if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})))
Assertion
Ref Expression
pwfseqlem2 ((π‘Œ ∈ Fin ∧ 𝑅 ∈ 𝑉) β†’ (π‘ŒπΉπ‘…) = (π»β€˜(cardβ€˜π‘Œ)))
Distinct variable groups:   𝑛,π‘Ÿ,𝑀,π‘₯,𝑧   𝐷,𝑛,𝑧   𝑀,𝐺   𝑀,𝐾   𝐻,π‘Ÿ,π‘₯,𝑧   πœ‘,𝑛,π‘Ÿ,π‘₯,𝑧   πœ“,𝑛,𝑧   𝐴,𝑛,π‘Ÿ,π‘₯,𝑧   𝑉,π‘Ÿ,π‘₯
Allowed substitution hints:   πœ‘(𝑀)   πœ“(π‘₯,𝑀,π‘Ÿ)   𝐴(𝑀)   𝐷(π‘₯,𝑀,π‘Ÿ)   𝑅(π‘₯,𝑧,𝑀,𝑛,π‘Ÿ)   𝐹(π‘₯,𝑧,𝑀,𝑛,π‘Ÿ)   𝐺(π‘₯,𝑧,𝑛,π‘Ÿ)   𝐻(𝑀,𝑛)   𝐾(π‘₯,𝑧,𝑛,π‘Ÿ)   𝑉(𝑧,𝑀,𝑛)   𝑋(π‘₯,𝑧,𝑀,𝑛,π‘Ÿ)   π‘Œ(π‘₯,𝑧,𝑀,𝑛,π‘Ÿ)

Proof of Theorem pwfseqlem2
Dummy variables π‘Ž 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7418 . . 3 (π‘Ž = π‘Œ β†’ (π‘ŽπΉπ‘ ) = (π‘ŒπΉπ‘ ))
2 2fveq3 6896 . . 3 (π‘Ž = π‘Œ β†’ (π»β€˜(cardβ€˜π‘Ž)) = (π»β€˜(cardβ€˜π‘Œ)))
31, 2eqeq12d 2748 . 2 (π‘Ž = π‘Œ β†’ ((π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž)) ↔ (π‘ŒπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Œ))))
4 oveq2 7419 . . 3 (𝑠 = 𝑅 β†’ (π‘ŒπΉπ‘ ) = (π‘ŒπΉπ‘…))
54eqeq1d 2734 . 2 (𝑠 = 𝑅 β†’ ((π‘ŒπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Œ)) ↔ (π‘ŒπΉπ‘…) = (π»β€˜(cardβ€˜π‘Œ))))
6 nfcv 2903 . . 3 β„²π‘₯π‘Ž
7 nfcv 2903 . . 3 β„²π‘Ÿπ‘Ž
8 nfcv 2903 . . 3 β„²π‘Ÿπ‘ 
9 pwfseqlem4.f . . . . . 6 𝐹 = (π‘₯ ∈ V, π‘Ÿ ∈ V ↦ if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})))
10 nfmpo1 7491 . . . . . 6 β„²π‘₯(π‘₯ ∈ V, π‘Ÿ ∈ V ↦ if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})))
119, 10nfcxfr 2901 . . . . 5 β„²π‘₯𝐹
12 nfcv 2903 . . . . 5 β„²π‘₯π‘Ÿ
136, 11, 12nfov 7441 . . . 4 β„²π‘₯(π‘ŽπΉπ‘Ÿ)
1413nfeq1 2918 . . 3 β„²π‘₯(π‘ŽπΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘Ž))
15 nfmpo2 7492 . . . . . 6 β„²π‘Ÿ(π‘₯ ∈ V, π‘Ÿ ∈ V ↦ if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})))
169, 15nfcxfr 2901 . . . . 5 β„²π‘ŸπΉ
177, 16, 8nfov 7441 . . . 4 β„²π‘Ÿ(π‘ŽπΉπ‘ )
1817nfeq1 2918 . . 3 β„²π‘Ÿ(π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž))
19 oveq1 7418 . . . 4 (π‘₯ = π‘Ž β†’ (π‘₯πΉπ‘Ÿ) = (π‘ŽπΉπ‘Ÿ))
20 2fveq3 6896 . . . 4 (π‘₯ = π‘Ž β†’ (π»β€˜(cardβ€˜π‘₯)) = (π»β€˜(cardβ€˜π‘Ž)))
2119, 20eqeq12d 2748 . . 3 (π‘₯ = π‘Ž β†’ ((π‘₯πΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘₯)) ↔ (π‘ŽπΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘Ž))))
22 oveq2 7419 . . . 4 (π‘Ÿ = 𝑠 β†’ (π‘ŽπΉπ‘Ÿ) = (π‘ŽπΉπ‘ ))
2322eqeq1d 2734 . . 3 (π‘Ÿ = 𝑠 β†’ ((π‘ŽπΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘Ž)) ↔ (π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž))))
24 vex 3478 . . . . . 6 π‘₯ ∈ V
25 vex 3478 . . . . . 6 π‘Ÿ ∈ V
26 fvex 6904 . . . . . . 7 (π»β€˜(cardβ€˜π‘₯)) ∈ V
27 fvex 6904 . . . . . . 7 (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯}) ∈ V
2826, 27ifex 4578 . . . . . 6 if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})) ∈ V
299ovmpt4g 7557 . . . . . 6 ((π‘₯ ∈ V ∧ π‘Ÿ ∈ V ∧ if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})) ∈ V) β†’ (π‘₯πΉπ‘Ÿ) = if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})))
3024, 25, 28, 29mp3an 1461 . . . . 5 (π‘₯πΉπ‘Ÿ) = if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯}))
31 iftrue 4534 . . . . 5 (π‘₯ ∈ Fin β†’ if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})) = (π»β€˜(cardβ€˜π‘₯)))
3230, 31eqtrid 2784 . . . 4 (π‘₯ ∈ Fin β†’ (π‘₯πΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘₯)))
3332adantr 481 . . 3 ((π‘₯ ∈ Fin ∧ π‘Ÿ ∈ 𝑉) β†’ (π‘₯πΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘₯)))
346, 7, 8, 14, 18, 21, 23, 33vtocl2gaf 3567 . 2 ((π‘Ž ∈ Fin ∧ 𝑠 ∈ 𝑉) β†’ (π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž)))
353, 5, 34vtocl2ga 3566 1 ((π‘Œ ∈ Fin ∧ 𝑅 ∈ 𝑉) β†’ (π‘ŒπΉπ‘…) = (π»β€˜(cardβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   βŠ† wss 3948  ifcif 4528  π’« cpw 4602  βˆ© cint 4950  βˆͺ ciun 4997   class class class wbr 5148   We wwe 5630   Γ— cxp 5674  β—‘ccnv 5675  ran crn 5677  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  Ο‰com 7857   ↑m cmap 8822   β‰Ό cdom 8939  Fincfn 8941  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  pwfseqlem4a  10658  pwfseqlem4  10659
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