Theorem bdayimaon 33896

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	⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

Proof of Theorem bdayimaon

Step	Hyp	Ref	Expression
1		bdayfo 33880	. . . . . 6 ⊢ bday : No –onto→On
2		fofun 6689	. . . . . 6 ⊢ ( bday : No –onto→On → Fun bday )
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6520	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V)
5	3, 4	mpan 687	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V)
6	5	uniexd 7595	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V)
7		imassrn 5980	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday
8		forn 6691	. . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On)
9	1, 8	ax-mp 5	. . . . 5 ⊢ ran bday = On
10	7, 9	sseqtri 3957	. . . 4 ⊢ ( bday “ 𝐴) ⊆ On
11		ssorduni 7629	. . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
12	10, 11	ax-mp 5	. . 3 ⊢ Ord ∪ ( bday “ 𝐴)
13	6, 12	jctil 520	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V))
14		elon2 6277	. . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V))
15		sucelon 7664	. . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ suc ∪ ( bday “ 𝐴) ∈ On)
16	14, 15	bitr3i 276	. 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On)
17	13, 16	sylib 217	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ran crn 5590   “ cima 5592  Ord word 6265  Oncon0 6266  suc csuc 6268  Fun wfun 6427  –onto→wfo 6431   No csur 33843   bday cbday 33845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-no 33846  df-bday 33848

This theorem is referenced by:  noetasuplem1  33936  noetainflem1  33940