Theorem bdayimaon 33204

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 33189	. . . . . 6 ⊢ bday : No –onto→On
2		fofun 6577	. . . . . 6 ⊢ ( bday : No –onto→On → Fun bday )
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6426	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V)
5	3, 4	mpan 688	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V)
6	5	uniexd 7454	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V)
7		imassrn 5926	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday
8		forn 6579	. . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On)
9	1, 8	ax-mp 5	. . . . 5 ⊢ ran bday = On
10	7, 9	sseqtri 3991	. . . 4 ⊢ ( bday “ 𝐴) ⊆ On
11		ssorduni 7486	. . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
12	10, 11	ax-mp 5	. . 3 ⊢ Ord ∪ ( bday “ 𝐴)
13	6, 12	jctil 522	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V))
14		elon2 6188	. . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V))
15		sucelon 7518	. . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ suc ∪ ( bday “ 𝐴) ∈ On)
16	14, 15	bitr3i 279	. 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On)
17	13, 16	sylib 220	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3486   ⊆ wss 3924  ∪ cuni 4824  ran crn 5542   “ cima 5544  Ord word 6176  Oncon0 6177  suc csuc 6179  Fun wfun 6335  –onto→wfo 6339   No csur 33154   bday cbday 33156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-pss 3942  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-tp 4558  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5446  df-eprel 5451  df-po 5460  df-so 5461  df-fr 5500  df-we 5502  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-ord 6180  df-on 6181  df-suc 6183  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-1o 8088  df-no 33157  df-bday 33159

This theorem is referenced by:  noetalem1  33224