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Theorem bdayimaon 31968
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 31953 . . . . . 6 bday : No onto→On
2 fofun 6154 . . . . . 6 ( bday : No onto→On → Fun bday )
31, 2ax-mp 5 . . . . 5 Fun bday
4 funimaexg 6013 . . . . 5 ((Fun bday 𝐴𝑉) → ( bday 𝐴) ∈ V)
53, 4mpan 706 . . . 4 (𝐴𝑉 → ( bday 𝐴) ∈ V)
6 uniexg 6997 . . . 4 (( bday 𝐴) ∈ V → ( bday 𝐴) ∈ V)
75, 6syl 17 . . 3 (𝐴𝑉 ( bday 𝐴) ∈ V)
8 imassrn 5512 . . . . 5 ( bday 𝐴) ⊆ ran bday
9 forn 6156 . . . . . 6 ( bday : No onto→On → ran bday = On)
101, 9ax-mp 5 . . . . 5 ran bday = On
118, 10sseqtri 3670 . . . 4 ( bday 𝐴) ⊆ On
12 ssorduni 7027 . . . 4 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
1311, 12ax-mp 5 . . 3 Ord ( bday 𝐴)
147, 13jctil 559 . 2 (𝐴𝑉 → (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
15 elon2 5772 . . 3 ( ( bday 𝐴) ∈ On ↔ (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
16 sucelon 7059 . . 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 1523  wcel 2030  Vcvv 3231  wss 3607   cuni 4468  ran crn 5144  cima 5146  Ord word 5760  Oncon0 5761  suc csuc 5763  Fun wfun 5920  ontowfo 5924   No csur 31918   bday cbday 31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-no 31921  df-bday 31923
This theorem is referenced by:  noetalem1  31988
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