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Theorem cantnfvalf 9688
Description: Lemma for cantnf 9716. The function appearing in cantnfval 9691 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
Hypothesis
Ref Expression
cantnfvalf.f 𝐹 = seqΟ‰((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷)), βˆ…)
Assertion
Ref Expression
cantnfvalf 𝐹:Ο‰βŸΆOn
Distinct variable groups:   𝑧,π‘˜,𝐴   𝐡,π‘˜,𝑧
Allowed substitution hints:   𝐢(𝑧,π‘˜)   𝐷(𝑧,π‘˜)   𝐹(𝑧,π‘˜)

Proof of Theorem cantnfvalf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfvalf.f . . 3 𝐹 = seqΟ‰((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷)), βˆ…)
21fnseqom 8474 . 2 𝐹 Fn Ο‰
3 nn0suc 7899 . . . 4 (π‘₯ ∈ Ο‰ β†’ (π‘₯ = βˆ… ∨ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = suc 𝑦))
4 fveq2 6892 . . . . . . 7 (π‘₯ = βˆ… β†’ (πΉβ€˜π‘₯) = (πΉβ€˜βˆ…))
5 0ex 5302 . . . . . . . 8 βˆ… ∈ V
61seqom0g 8475 . . . . . . . 8 (βˆ… ∈ V β†’ (πΉβ€˜βˆ…) = βˆ…)
75, 6ax-mp 5 . . . . . . 7 (πΉβ€˜βˆ…) = βˆ…
84, 7eqtrdi 2781 . . . . . 6 (π‘₯ = βˆ… β†’ (πΉβ€˜π‘₯) = βˆ…)
9 0elon 6418 . . . . . 6 βˆ… ∈ On
108, 9eqeltrdi 2833 . . . . 5 (π‘₯ = βˆ… β†’ (πΉβ€˜π‘₯) ∈ On)
111seqomsuc 8476 . . . . . . . . 9 (𝑦 ∈ Ο‰ β†’ (πΉβ€˜suc 𝑦) = (𝑦(π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))(πΉβ€˜π‘¦)))
12 df-ov 7419 . . . . . . . . 9 (𝑦(π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))(πΉβ€˜π‘¦)) = ((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))β€˜βŸ¨π‘¦, (πΉβ€˜π‘¦)⟩)
1311, 12eqtrdi 2781 . . . . . . . 8 (𝑦 ∈ Ο‰ β†’ (πΉβ€˜suc 𝑦) = ((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))β€˜βŸ¨π‘¦, (πΉβ€˜π‘¦)⟩))
14 df-ov 7419 . . . . . . . . . . . 12 (𝐢 +o 𝐷) = ( +o β€˜βŸ¨πΆ, 𝐷⟩)
15 fnoa 8527 . . . . . . . . . . . . . 14 +o Fn (On Γ— On)
16 oacl 8554 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ +o 𝑦) ∈ On)
1716rgen2 3188 . . . . . . . . . . . . . 14 βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ +o 𝑦) ∈ On
18 ffnov 7544 . . . . . . . . . . . . . 14 ( +o :(On Γ— On)⟢On ↔ ( +o Fn (On Γ— On) ∧ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ +o 𝑦) ∈ On))
1915, 17, 18mpbir2an 709 . . . . . . . . . . . . 13 +o :(On Γ— On)⟢On
2019, 9f0cli 7103 . . . . . . . . . . . 12 ( +o β€˜βŸ¨πΆ, 𝐷⟩) ∈ On
2114, 20eqeltri 2821 . . . . . . . . . . 11 (𝐢 +o 𝐷) ∈ On
2221rgen2w 3056 . . . . . . . . . 10 βˆ€π‘˜ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 (𝐢 +o 𝐷) ∈ On
23 eqid 2725 . . . . . . . . . . 11 (π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷)) = (π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))
2423fmpo 8070 . . . . . . . . . 10 (βˆ€π‘˜ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 (𝐢 +o 𝐷) ∈ On ↔ (π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷)):(𝐴 Γ— 𝐡)⟢On)
2522, 24mpbi 229 . . . . . . . . 9 (π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷)):(𝐴 Γ— 𝐡)⟢On
2625, 9f0cli 7103 . . . . . . . 8 ((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))β€˜βŸ¨π‘¦, (πΉβ€˜π‘¦)⟩) ∈ On
2713, 26eqeltrdi 2833 . . . . . . 7 (𝑦 ∈ Ο‰ β†’ (πΉβ€˜suc 𝑦) ∈ On)
28 fveq2 6892 . . . . . . . 8 (π‘₯ = suc 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜suc 𝑦))
2928eleq1d 2810 . . . . . . 7 (π‘₯ = suc 𝑦 β†’ ((πΉβ€˜π‘₯) ∈ On ↔ (πΉβ€˜suc 𝑦) ∈ On))
3027, 29syl5ibrcom 246 . . . . . 6 (𝑦 ∈ Ο‰ β†’ (π‘₯ = suc 𝑦 β†’ (πΉβ€˜π‘₯) ∈ On))
3130rexlimiv 3138 . . . . 5 (βˆƒπ‘¦ ∈ Ο‰ π‘₯ = suc 𝑦 β†’ (πΉβ€˜π‘₯) ∈ On)
3210, 31jaoi 855 . . . 4 ((π‘₯ = βˆ… ∨ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = suc 𝑦) β†’ (πΉβ€˜π‘₯) ∈ On)
333, 32syl 17 . . 3 (π‘₯ ∈ Ο‰ β†’ (πΉβ€˜π‘₯) ∈ On)
3433rgen 3053 . 2 βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜π‘₯) ∈ On
35 ffnfv 7124 . 2 (𝐹:Ο‰βŸΆOn ↔ (𝐹 Fn Ο‰ ∧ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜π‘₯) ∈ On))
362, 34, 35mpbir2an 709 1 𝐹:Ο‰βŸΆOn
Colors of variables: wff setvar class
Syntax hints:   ∨ wo 845   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060  Vcvv 3463  βˆ…c0 4318  βŸ¨cop 4630   Γ— cxp 5670  Oncon0 6364  suc csuc 6366   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  Ο‰com 7868  seqΟ‰cseqom 8466   +o coa 8482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seqom 8467  df-oadd 8489
This theorem is referenced by:  cantnfval2  9692  cantnfle  9694  cantnflt  9695  cantnflem1d  9711  cantnflem1  9712  cnfcomlem  9722
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