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	⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

Proof of Theorem bdayimaon

Step	Hyp	Ref	Expression
1		bdayfo 32421	. . . . . 6 ⊢ bday : No –onto→On
2		fofun 6369	. . . . . 6 ⊢ ( bday : No –onto→On → Fun bday )
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6222	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V)
5	3, 4	mpan 680	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V)
6		uniexg 7234	. . . 4 ⊢ (( bday “ 𝐴) ∈ V → ∪ ( bday “ 𝐴) ∈ V)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V)
8		imassrn 5733	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday
9		forn 6371	. . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On)
10	1, 9	ax-mp 5	. . . . 5 ⊢ ran bday = On
11	8, 10	sseqtri 3856	. . . 4 ⊢ ( bday “ 𝐴) ⊆ On
12		ssorduni 7265	. . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
13	11, 12	ax-mp 5	. . 3 ⊢ Ord ∪ ( bday “ 𝐴)
14	7, 13	jctil 515	. 2 ⊢ (𝐴 ∈ 𝑉 → (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)
17	15, 16	bitr3i 269	. 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On)
18	14, 17	sylib 210	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

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  –onto→wfo 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