Theorem nfoi 9273

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 9269	. 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 5565	. . . 4 ⊢ Ⅎ𝑥 𝑅 We 𝐴
5	2, 3	nfse 5564	. . . 4 ⊢ Ⅎ𝑥 𝑅 Se 𝐴
6	4, 5	nfan 1902	. . 3 ⊢ Ⅎ𝑥(𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
7		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥V
8		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran ℎ
9		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑗
10		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
11	9, 2, 10	nfbr 5121	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑗𝑅𝑤
12	8, 11	nfralw 3151	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤
13	12, 3	nfrabw 3318	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
14		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑢
15		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
16	14, 2, 15	nfbr 5121	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
17	16	nfn 1860	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
18	13, 17	nfralw 3151	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
19	18, 13	nfriota 7245	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
20	7, 19	nfmpt 5181	. . . . 5 ⊢ Ⅎ𝑥(ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
21	20	nfrecs 8206	. . . 4 ⊢ Ⅎ𝑥recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
23	21, 22	nfima 5977	. . . . . . 7 ⊢ Ⅎ𝑥(recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
25		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
26	24, 2, 25	nfbr 5121	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧𝑅𝑡
27	23, 26	nfralw 3151	. . . . . 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	nfrex 3242	. . . . 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 2907	. . . . 5 ⊢ Ⅎ𝑥On
30	28, 29	nfrabw 3318	. . . 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 5893	. . 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 2907	. . 3 ⊢ Ⅎ𝑥∅
33	6, 31, 32	nfif 4489	. 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 2905	1 ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 396  Ⅎwnfc 2887  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432  ∅c0 4256  ifcif 4459   class class class wbr 5074   ↦ cmpt 5157   Se wse 5542   We wwe 5543  ran crn 5590   ↾ cres 5591   “ cima 5592  Oncon0 6266  ℩crio 7231  recscrecs 8201  OrdIsocoi 9268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fv 6441  df-riota 7232  df-ov 7278  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-oi 9269

This theorem is referenced by:  hsmexlem2  10183