Theorem nfoi 9425

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 9421	. 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 5598	. . . 4 ⊢ Ⅎ𝑥 𝑅 We 𝐴
5	2, 3	nfse 5597	. . . 4 ⊢ Ⅎ𝑥 𝑅 Se 𝐴
6	4, 5	nfan 1899	. . 3 ⊢ Ⅎ𝑥(𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
7		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥V
8		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran ℎ
9		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑗
10		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
11	9, 2, 10	nfbr 5142	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑗𝑅𝑤
12	8, 11	nfralw 3277	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤
13	12, 3	nfrabw 3434	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
14		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑢
15		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
16	14, 2, 15	nfbr 5142	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
17	16	nfn 1857	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
18	13, 17	nfralw 3277	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
19	18, 13	nfriota 7322	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
20	7, 19	nfmpt 5193	. . . . 5 ⊢ Ⅎ𝑥(ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
21	20	nfrecs 8304	. . . 4 ⊢ Ⅎ𝑥recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
23	21, 22	nfima 6023	. . . . . . 7 ⊢ Ⅎ𝑥(recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
25		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
26	24, 2, 25	nfbr 5142	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧𝑅𝑡
27	23, 26	nfralw 3277	. . . . . 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 3278	. . . . 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 2891	. . . . 5 ⊢ Ⅎ𝑥On
30	28, 29	nfrabw 3434	. . . 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 5936	. . 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 2891	. . 3 ⊢ Ⅎ𝑥∅
33	6, 31, 32	nfif 4509	. 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 2889	1 ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053  {crab 3396  Vcvv 3438  ∅c0 4286  ifcif 4478   class class class wbr 5095   ↦ cmpt 5176   Se wse 5574   We wwe 5575  ran crn 5624   ↾ cres 5625   “ cima 5626  Oncon0 6311  ℩crio 7309  recscrecs 8300  OrdIsocoi 9420

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-iota 6442  df-fv 6494  df-riota 7310  df-ov 7356  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-oi 9421

This theorem is referenced by:  hsmexlem2  10340