Theorem nfoi 9400

Description: Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfoi.1	⊢ Ⅎ𝑥𝑅
nfoi.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfoi	⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)

Proof of Theorem nfoi

Dummy variables ℎ 𝑎 𝑗 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oi 9396	. 2 ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑎 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}), ∅)
2		nfoi.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
3		nfoi.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfwe 5591	. . . 4 ⊢ Ⅎ𝑥 𝑅 We 𝐴
5	2, 3	nfse 5590	. . . 4 ⊢ Ⅎ𝑥 𝑅 Se 𝐴
6	4, 5	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
7		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥V
8		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran ℎ
9		nfcv 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑗
10		nfcv 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
11	9, 2, 10	nfbr 5138	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑗𝑅𝑤
12	8, 11	nfralw 3279	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤
13	12, 3	nfrabw 3432	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
14		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑢
15		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
16	14, 2, 15	nfbr 5138	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
17	16	nfn 1858	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
18	13, 17	nfralw 3279	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
19	18, 13	nfriota 7315	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
20	7, 19	nfmpt 5189	. . . . 5 ⊢ Ⅎ𝑥(ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
21	20	nfrecs 8294	. . . 4 ⊢ Ⅎ𝑥recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
23	21, 22	nfima 6017	. . . . . . 7 ⊢ Ⅎ𝑥(recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
25		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
26	24, 2, 25	nfbr 5138	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧𝑅𝑡
27	23, 26	nfralw 3279	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
28	3, 27	nfrexw 3280	. . . . 5 ⊢ Ⅎ𝑥∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
29		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥On
30	28, 29	nfrabw 3432	. . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}
31	21, 30	nfres 5930	. . 3 ⊢ Ⅎ𝑥(recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑎 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡})
32		nfcv 2894	. . 3 ⊢ Ⅎ𝑥∅
33	6, 31, 32	nfif 4506	. 2 ⊢ Ⅎ𝑥if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑎 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}), ∅)
34	1, 33	nfcxfr 2892	1 ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395  Ⅎwnfc 2879  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436  ∅c0 4283  ifcif 4475   class class class wbr 5091   ↦ cmpt 5172   Se wse 5567   We wwe 5568  ran crn 5617   ↾ cres 5618   “ cima 5619  Oncon0 6306  ℩crio 7302  recscrecs 8290  OrdIsocoi 9395

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-iota 6437  df-fv 6489  df-riota 7303  df-ov 7349  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-oi 9396

This theorem is referenced by:  hsmexlem2  10318