Theorem nfoi 9533

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 9529	. 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 5634	. . . 4 ⊢ Ⅎ𝑥 𝑅 We 𝐴
5	2, 3	nfse 5633	. . . 4 ⊢ Ⅎ𝑥 𝑅 Se 𝐴
6	4, 5	nfan 1899	. . 3 ⊢ Ⅎ𝑥(𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
7		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥V
8		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran ℎ
9		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑗
10		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
11	9, 2, 10	nfbr 5171	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑗𝑅𝑤
12	8, 11	nfralw 3295	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤
13	12, 3	nfrabw 3459	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
14		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑢
15		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
16	14, 2, 15	nfbr 5171	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
17	16	nfn 1857	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
18	13, 17	nfralw 3295	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
19	18, 13	nfriota 7379	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
20	7, 19	nfmpt 5224	. . . . 5 ⊢ Ⅎ𝑥(ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
21	20	nfrecs 8394	. . . 4 ⊢ Ⅎ𝑥recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
23	21, 22	nfima 6060	. . . . . . 7 ⊢ Ⅎ𝑥(recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
25		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
26	24, 2, 25	nfbr 5171	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧𝑅𝑡
27	23, 26	nfralw 3295	. . . . . 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 3297	. . . . 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 2899	. . . . 5 ⊢ Ⅎ𝑥On
30	28, 29	nfrabw 3459	. . . 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 5973	. . 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 2899	. . 3 ⊢ Ⅎ𝑥∅
33	6, 31, 32	nfif 4536	. 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 2897	1 ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464  ∅c0 4313  ifcif 4505   class class class wbr 5124   ↦ cmpt 5206   Se wse 5609   We wwe 5610  ran crn 5660   ↾ cres 5661   “ cima 5662  Oncon0 6357  ℩crio 7366  recscrecs 8389  OrdIsocoi 9528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-iota 6489  df-fv 6544  df-riota 7367  df-ov 7413  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-oi 9529

This theorem is referenced by:  hsmexlem2  10446