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Theorem bdayimaon 32145
Description: Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
Assertion
Ref Expression
bdayimaon (𝐴𝑉 → suc ( bday 𝐴) ∈ On)

Proof of Theorem bdayimaon
StepHypRef Expression
1 bdayfo 32130 . . . . . 6 bday : No onto→On
2 fofun 6273 . . . . . 6 ( bday : No onto→On → Fun bday )
31, 2ax-mp 5 . . . . 5 Fun bday
4 funimaexg 6132 . . . . 5 ((Fun bday 𝐴𝑉) → ( bday 𝐴) ∈ V)
53, 4mpan 708 . . . 4 (𝐴𝑉 → ( bday 𝐴) ∈ V)
6 uniexg 7116 . . . 4 (( bday 𝐴) ∈ V → ( bday 𝐴) ∈ V)
75, 6syl 17 . . 3 (𝐴𝑉 ( bday 𝐴) ∈ V)
8 imassrn 5631 . . . . 5 ( bday 𝐴) ⊆ ran bday
9 forn 6275 . . . . . 6 ( bday : No onto→On → ran bday = On)
101, 9ax-mp 5 . . . . 5 ran bday = On
118, 10sseqtri 3774 . . . 4 ( bday 𝐴) ⊆ On
12 ssorduni 7146 . . . 4 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
1311, 12ax-mp 5 . . 3 Ord ( bday 𝐴)
147, 13jctil 561 . 2 (𝐴𝑉 → (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
15 elon2 5891 . . 3 ( ( bday 𝐴) ∈ On ↔ (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
16 sucelon 7178 . . 3 ( ( bday 𝐴) ∈ On ↔ suc ( bday 𝐴) ∈ On)
1715, 16bitr3i 266 . 2 ((Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V) ↔ suc ( bday 𝐴) ∈ On)
1814, 17sylib 208 1 (𝐴𝑉 → suc ( bday 𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  Vcvv 3336  wss 3711   cuni 4584  ran crn 5263  cima 5265  Ord word 5879  Oncon0 5880  suc csuc 5882  Fun wfun 6039  ontowfo 6043   No csur 32095   bday cbday 32097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-ord 5883  df-on 5884  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-1o 7725  df-no 32098  df-bday 32100
This theorem is referenced by:  noetalem1  32165
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