Theorem bdayimaon 27756

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 27740	. . . . . 6 ⊢ bday : No –onto→On
2		fofun 6835	. . . . . 6 ⊢ ( bday : No –onto→On → Fun bday )
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6664	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V)
5	3, 4	mpan 689	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V)
6	5	uniexd 7777	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V)
7		imassrn 6100	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday
8		forn 6837	. . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On)
9	1, 8	ax-mp 5	. . . . 5 ⊢ ran bday = On
10	7, 9	sseqtri 4045	. . . 4 ⊢ ( bday “ 𝐴) ⊆ On
11		ssorduni 7814	. . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
12	10, 11	ax-mp 5	. . 3 ⊢ Ord ∪ ( bday “ 𝐴)
13	6, 12	jctil 519	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V))
14		elon2 6406	. . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V))
15		onsucb 7853	. . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ suc ∪ ( bday “ 𝐴) ∈ On)
16	14, 15	bitr3i 277	. 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On)
17	13, 16	sylib 218	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∪ cuni 4931  ran crn 5701   “ cima 5703  Ord word 6394  Oncon0 6395  suc csuc 6397  Fun wfun 6567  –onto→wfo 6571   No csur 27702   bday cbday 27704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-1o 8522  df-no 27705  df-bday 27707

This theorem is referenced by:  noetasuplem1  27796  noetainflem1  27800