Theorem bdayimaon 28077

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 28061	. . . . . 6 ⊢ bday:No⟶ontoOn
2		fofun 6816	. . . . . 6 ⊢ (bday:No⟶ontoOn → Fun bday)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun bday
4		funimaexg 6645	. . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → (bday “ 𝐴) ∈ V)
5	3, 4	mpan 703	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (bday “ 𝐴) ∈ V)
6	5	uniexd 7780	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ (bday “ 𝐴) ∈ V)
7		imassrn 6209	. . . . 5 ⊢ (bday “ 𝐴) ⊆ ran bday
8		forn 6818	. . . . . 6 ⊢ (bday:No⟶ontoOn → ran bday = On)
9	1, 8	ax-mp 5	. . . . 5 ⊢ ran bday = On
10	7, 9	sseqtri 3961	. . . 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 529	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ (bday “ 𝐴) ∧ ∪ (bday “ 𝐴) ∈ V))
14		elon2 6388	. . 3 ⊢ (∪ (bday “ 𝐴) ∈ On ↔ (Ord ∪ (bday “ 𝐴) ∧ ∪ (bday “ 𝐴) ∈ V))
15		onsucb 7849	. . 3 ⊢ (∪ (bday “ 𝐴) ∈ On ↔ suc ∪ (bday “ 𝐴) ∈ On)
16	14, 15	bitr3i 280	. 2 ⊢ ((Ord ∪ (bday “ 𝐴) ∧ ∪ (bday “ 𝐴) ∈ V) ↔ suc ∪ (bday “ 𝐴) ∈ On)
17	13, 16	sylib 221	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ (bday “ 𝐴) ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∪ cuni 4870  ran crn 5656   “ cima 5658  Ord word 6376  Oncon0 6377  suc csuc 6379  Fun wfun 6549  ⟶ontowfo 6553  Nocsur 28024  bdaycbday 28026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-suc 6383  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-1o 8501  df-no 28027  df-bday 28029

This theorem is used by:  noetasuplem1  28117  noetainflem1  28121