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Theorem pwfseqlem2 10603
Description: Lemma for pwfseq 10608. (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 7368 . . 3 (π‘Ž = π‘Œ β†’ (π‘ŽπΉπ‘ ) = (π‘ŒπΉπ‘ ))
2 2fveq3 6851 . . 3 (π‘Ž = π‘Œ β†’ (π»β€˜(cardβ€˜π‘Ž)) = (π»β€˜(cardβ€˜π‘Œ)))
31, 2eqeq12d 2749 . 2 (π‘Ž = π‘Œ β†’ ((π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž)) ↔ (π‘ŒπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Œ))))
4 oveq2 7369 . . 3 (𝑠 = 𝑅 β†’ (π‘ŒπΉπ‘ ) = (π‘ŒπΉπ‘…))
54eqeq1d 2735 . 2 (𝑠 = 𝑅 β†’ ((π‘ŒπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Œ)) ↔ (π‘ŒπΉπ‘…) = (π»β€˜(cardβ€˜π‘Œ))))
6 nfcv 2904 . . 3 β„²π‘₯π‘Ž
7 nfcv 2904 . . 3 β„²π‘Ÿπ‘Ž
8 nfcv 2904 . . 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 7441 . . . . . 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 2902 . . . . 5 β„²π‘₯𝐹
12 nfcv 2904 . . . . 5 β„²π‘₯π‘Ÿ
136, 11, 12nfov 7391 . . . 4 β„²π‘₯(π‘ŽπΉπ‘Ÿ)
1413nfeq1 2919 . . 3 β„²π‘₯(π‘ŽπΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘Ž))
15 nfmpo2 7442 . . . . . 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 2902 . . . . 5 β„²π‘ŸπΉ
177, 16, 8nfov 7391 . . . 4 β„²π‘Ÿ(π‘ŽπΉπ‘ )
1817nfeq1 2919 . . 3 β„²π‘Ÿ(π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž))
19 oveq1 7368 . . . 4 (π‘₯ = π‘Ž β†’ (π‘₯πΉπ‘Ÿ) = (π‘ŽπΉπ‘Ÿ))
20 2fveq3 6851 . . . 4 (π‘₯ = π‘Ž β†’ (π»β€˜(cardβ€˜π‘₯)) = (π»β€˜(cardβ€˜π‘Ž)))
2119, 20eqeq12d 2749 . . 3 (π‘₯ = π‘Ž β†’ ((π‘₯πΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘₯)) ↔ (π‘ŽπΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘Ž))))
22 oveq2 7369 . . . 4 (π‘Ÿ = 𝑠 β†’ (π‘ŽπΉπ‘Ÿ) = (π‘ŽπΉπ‘ ))
2322eqeq1d 2735 . . 3 (π‘Ÿ = 𝑠 β†’ ((π‘ŽπΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘Ž)) ↔ (π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž))))
24 vex 3451 . . . . . 6 π‘₯ ∈ V
25 vex 3451 . . . . . 6 π‘Ÿ ∈ V
26 fvex 6859 . . . . . . 7 (π»β€˜(cardβ€˜π‘₯)) ∈ V
27 fvex 6859 . . . . . . 7 (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯}) ∈ V
2826, 27ifex 4540 . . . . . 6 if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯})) ∈ V
299ovmpt4g 7506 . . . . . 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 1462 . . . . 5 (π‘₯πΉπ‘Ÿ) = if(π‘₯ ∈ Fin, (π»β€˜(cardβ€˜π‘₯)), (π·β€˜βˆ© {𝑧 ∈ Ο‰ ∣ Β¬ (π·β€˜π‘§) ∈ π‘₯}))
31 iftrue 4496 . . . . 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 2785 . . . 4 (π‘₯ ∈ Fin β†’ (π‘₯πΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘₯)))
3332adantr 482 . . 3 ((π‘₯ ∈ Fin ∧ π‘Ÿ ∈ 𝑉) β†’ (π‘₯πΉπ‘Ÿ) = (π»β€˜(cardβ€˜π‘₯)))
346, 7, 8, 14, 18, 21, 23, 33vtocl2gaf 3538 . 2 ((π‘Ž ∈ Fin ∧ 𝑠 ∈ 𝑉) β†’ (π‘ŽπΉπ‘ ) = (π»β€˜(cardβ€˜π‘Ž)))
353, 5, 34vtocl2ga 3537 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3447   βŠ† wss 3914  ifcif 4490  π’« cpw 4564  βˆ© cint 4911  βˆͺ ciun 4958   class class class wbr 5109   We wwe 5591   Γ— cxp 5635  β—‘ccnv 5636  ran crn 5638  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Ο‰com 7806   ↑m cmap 8771   β‰Ό cdom 8887  Fincfn 8889  cardccrd 9879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366
This theorem is referenced by:  pwfseqlem4a  10605  pwfseqlem4  10606
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