MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfvalf Structured version   Visualization version   GIF version

Theorem cantnfvalf 9609
Description: Lemma for cantnf 9637. The function appearing in cantnfval 9612 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 8405 . 2 𝐹 Fn Ο‰
3 nn0suc 7836 . . . 4 (π‘₯ ∈ Ο‰ β†’ (π‘₯ = βˆ… ∨ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = suc 𝑦))
4 fveq2 6846 . . . . . . 7 (π‘₯ = βˆ… β†’ (πΉβ€˜π‘₯) = (πΉβ€˜βˆ…))
5 0ex 5268 . . . . . . . 8 βˆ… ∈ V
61seqom0g 8406 . . . . . . . 8 (βˆ… ∈ V β†’ (πΉβ€˜βˆ…) = βˆ…)
75, 6ax-mp 5 . . . . . . 7 (πΉβ€˜βˆ…) = βˆ…
84, 7eqtrdi 2789 . . . . . 6 (π‘₯ = βˆ… β†’ (πΉβ€˜π‘₯) = βˆ…)
9 0elon 6375 . . . . . 6 βˆ… ∈ On
108, 9eqeltrdi 2842 . . . . 5 (π‘₯ = βˆ… β†’ (πΉβ€˜π‘₯) ∈ On)
111seqomsuc 8407 . . . . . . . . 9 (𝑦 ∈ Ο‰ β†’ (πΉβ€˜suc 𝑦) = (𝑦(π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))(πΉβ€˜π‘¦)))
12 df-ov 7364 . . . . . . . . 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 2789 . . . . . . . 8 (𝑦 ∈ Ο‰ β†’ (πΉβ€˜suc 𝑦) = ((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))β€˜βŸ¨π‘¦, (πΉβ€˜π‘¦)⟩))
14 df-ov 7364 . . . . . . . . . . . 12 (𝐢 +o 𝐷) = ( +o β€˜βŸ¨πΆ, 𝐷⟩)
15 fnoa 8458 . . . . . . . . . . . . . 14 +o Fn (On Γ— On)
16 oacl 8485 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ +o 𝑦) ∈ On)
1716rgen2 3191 . . . . . . . . . . . . . 14 βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ +o 𝑦) ∈ On
18 ffnov 7487 . . . . . . . . . . . . . 14 ( +o :(On Γ— On)⟢On ↔ ( +o Fn (On Γ— On) ∧ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ +o 𝑦) ∈ On))
1915, 17, 18mpbir2an 710 . . . . . . . . . . . . 13 +o :(On Γ— On)⟢On
2019, 9f0cli 7052 . . . . . . . . . . . 12 ( +o β€˜βŸ¨πΆ, 𝐷⟩) ∈ On
2114, 20eqeltri 2830 . . . . . . . . . . 11 (𝐢 +o 𝐷) ∈ On
2221rgen2w 3066 . . . . . . . . . 10 βˆ€π‘˜ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 (𝐢 +o 𝐷) ∈ On
23 eqid 2733 . . . . . . . . . . 11 (π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷)) = (π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))
2423fmpo 8004 . . . . . . . . . 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 7052 . . . . . . . 8 ((π‘˜ ∈ 𝐴, 𝑧 ∈ 𝐡 ↦ (𝐢 +o 𝐷))β€˜βŸ¨π‘¦, (πΉβ€˜π‘¦)⟩) ∈ On
2713, 26eqeltrdi 2842 . . . . . . 7 (𝑦 ∈ Ο‰ β†’ (πΉβ€˜suc 𝑦) ∈ On)
28 fveq2 6846 . . . . . . . 8 (π‘₯ = suc 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜suc 𝑦))
2928eleq1d 2819 . . . . . . 7 (π‘₯ = suc 𝑦 β†’ ((πΉβ€˜π‘₯) ∈ On ↔ (πΉβ€˜suc 𝑦) ∈ On))
3027, 29syl5ibrcom 247 . . . . . 6 (𝑦 ∈ Ο‰ β†’ (π‘₯ = suc 𝑦 β†’ (πΉβ€˜π‘₯) ∈ On))
3130rexlimiv 3142 . . . . 5 (βˆƒπ‘¦ ∈ Ο‰ π‘₯ = suc 𝑦 β†’ (πΉβ€˜π‘₯) ∈ On)
3210, 31jaoi 856 . . . 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 3063 . 2 βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜π‘₯) ∈ On
35 ffnfv 7070 . 2 (𝐹:Ο‰βŸΆOn ↔ (𝐹 Fn Ο‰ ∧ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜π‘₯) ∈ On))
362, 34, 35mpbir2an 710 1 𝐹:Ο‰βŸΆOn
Colors of variables: wff setvar class
Syntax hints:   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447  βˆ…c0 4286  βŸ¨cop 4596   Γ— cxp 5635  Oncon0 6321  suc csuc 6323   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Ο‰com 7806  seqΟ‰cseqom 8397   +o coa 8413
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seqom 8398  df-oadd 8420
This theorem is referenced by:  cantnfval2  9613  cantnfle  9615  cantnflt  9616  cantnflem1d  9632  cantnflem1  9633  cnfcomlem  9643
  Copyright terms: Public domain W3C validator