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Theorem bdayimaon 27125
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 27109 . . . . . 6 bday : No onto→On
2 fofun 6794 . . . . . 6 ( bday : No onto→On → Fun bday )
31, 2ax-mp 5 . . . . 5 Fun bday
4 funimaexg 6624 . . . . 5 ((Fun bday 𝐴𝑉) → ( bday 𝐴) ∈ V)
53, 4mpan 688 . . . 4 (𝐴𝑉 → ( bday 𝐴) ∈ V)
65uniexd 7716 . . 3 (𝐴𝑉 ( bday 𝐴) ∈ V)
7 imassrn 6061 . . . . 5 ( bday 𝐴) ⊆ ran bday
8 forn 6796 . . . . . 6 ( bday : No onto→On → ran bday = On)
91, 8ax-mp 5 . . . . 5 ran bday = On
107, 9sseqtri 4015 . . . 4 ( bday 𝐴) ⊆ On
11 ssorduni 7750 . . . 4 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
1210, 11ax-mp 5 . . 3 Ord ( bday 𝐴)
136, 12jctil 520 . 2 (𝐴𝑉 → (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
14 elon2 6365 . . 3 ( ( bday 𝐴) ∈ On ↔ (Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V))
15 onsucb 7789 . . 3 ( ( bday 𝐴) ∈ On ↔ suc ( bday 𝐴) ∈ On)
1614, 15bitr3i 276 . 2 ((Ord ( bday 𝐴) ∧ ( bday 𝐴) ∈ V) ↔ suc ( bday 𝐴) ∈ On)
1713, 16sylib 217 1 (𝐴𝑉 → suc ( bday 𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  wss 3945   cuni 4902  ran crn 5671  cima 5673  Ord word 6353  Oncon0 6354  suc csuc 6356  Fun wfun 6527  ontowfo 6531   No csur 27072   bday cbday 27074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-1o 8450  df-no 27075  df-bday 27077
This theorem is referenced by:  noetasuplem1  27165  noetainflem1  27169
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