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	⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

Proof of Theorem bdayimaon

Step	Hyp	Ref	Expression
1		bdayfo 31953	. . . . . 6 ⊢ bday : No –onto→On
2		fofun 6154	. . . . . 6 ⊢ ( bday : No –onto→On → Fun bday )
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6013	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V)
5	3, 4	mpan 706	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V)
6		uniexg 6997	. . . 4 ⊢ (( bday “ 𝐴) ∈ V → ∪ ( bday “ 𝐴) ∈ V)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V)
8		imassrn 5512	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday
9		forn 6156	. . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On)
10	1, 9	ax-mp 5	. . . . 5 ⊢ ran bday = On
11	8, 10	sseqtri 3670	. . . 4 ⊢ ( bday “ 𝐴) ⊆ On
12		ssorduni 7027	. . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
13	11, 12	ax-mp 5	. . 3 ⊢ Ord ∪ ( bday “ 𝐴)
14	7, 13	jctil 559	. 2 ⊢ (𝐴 ∈ 𝑉 → (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)
17	15, 16	bitr3i 266	. 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On)
18	14, 17	sylib 208	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

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