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Theorem bdayimaon 32436
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 32421 . . . . . 6 bday : No onto→On
2 fofun 6369 . . . . . 6 ( bday : No onto→On → Fun bday )
31, 2ax-mp 5 . . . . 5 Fun bday
4 funimaexg 6222 . . . . 5 ((Fun bday 𝐴𝑉) → ( bday 𝐴) ∈ V)
53, 4mpan 680 . . . 4 (𝐴𝑉 → ( bday 𝐴) ∈ V)
6 uniexg 7234 . . . 4 (( bday 𝐴) ∈ V → ( bday 𝐴) ∈ V)
75, 6syl 17 . . 3 (𝐴𝑉 ( bday 𝐴) ∈ V)
8 imassrn 5733 . . . . 5 ( bday 𝐴) ⊆ ran bday
9 forn 6371 . . . . . 6 ( bday : No onto→On → ran bday = On)
101, 9ax-mp 5 . . . . 5 ran bday = On
118, 10sseqtri 3856 . . . 4 ( bday 𝐴) ⊆ On
12 ssorduni 7265 . . . 4 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
1311, 12ax-mp 5 . . 3 Ord ( bday 𝐴)
147, 13jctil 515 . 2 (𝐴𝑉 → (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
15 elon2 5989 . . 3 ( ( bday 𝐴) ∈ On ↔ (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
16 sucelon 7297 . . 3 ( ( bday 𝐴) ∈ On ↔ suc ( bday 𝐴) ∈ On)
1715, 16bitr3i 269 . 2 ((Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V) ↔ suc ( bday 𝐴) ∈ On)
1814, 17sylib 210 1 (𝐴𝑉 → suc ( bday 𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  wss 3792   cuni 4673  ran crn 5358  cima 5360  Ord word 5977  Oncon0 5978  suc csuc 5980  Fun wfun 6131  ontowfo 6135   No csur 32386   bday cbday 32388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-ord 5981  df-on 5982  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-1o 7845  df-no 32389  df-bday 32391
This theorem is referenced by:  noetalem1  32456
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