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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  –onto→wfo 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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