Theorem nfoi 9455

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 9451	. 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 5618	. . . 4 ⊢ Ⅎ𝑥 𝑅 We 𝐴
5	2, 3	nfse 5617	. . . 4 ⊢ Ⅎ𝑥 𝑅 Se 𝐴
6	4, 5	nfan 1918	. . 3 ⊢ Ⅎ𝑥(𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
7		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥V
8		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran ℎ
9		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑗
10		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
11	9, 2, 10	nfbr 5144	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑗𝑅𝑤
12	8, 11	nfralw 3308	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤
13	12, 3	nfrabw 3450	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
14		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑢
15		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
16	14, 2, 15	nfbr 5144	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
17	16	nfn 1876	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
18	13, 17	nfralw 3308	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
19	18, 13	nfriota 7359	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
20	7, 19	nfmpt 5195	. . . . 5 ⊢ Ⅎ𝑥(ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
21	20	nfrecs 8338	. . . 4 ⊢ Ⅎ𝑥recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
23	21, 22	nfima 6052	. . . . . . 7 ⊢ Ⅎ𝑥(recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
25		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
26	24, 2, 25	nfbr 5144	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧𝑅𝑡
27	23, 26	nfralw 3308	. . . . . 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 3309	. . . . 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 2923	. . . . 5 ⊢ Ⅎ𝑥On
30	28, 29	nfrabw 3450	. . . 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 5963	. . 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 2923	. . 3 ⊢ Ⅎ𝑥∅
33	6, 31, 32	nfif 4508	. 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 2921	1 ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 399  Ⅎwnfc 2908  ∀wral 3075  ∃wrex 3085  {crab 3413  Vcvv 3453  ∅c0 4283  ifcif 4477   class class class wbr 5097   ↦ cmpt 5178   Se wse 5594   We wwe 5595  ran crn 5644   ↾ cres 5645   “ cima 5646  Oncon0 6340  ℩crio 7346  recscrecs 8334  OrdIsocoi 9450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-iota 6471  df-fv 6523  df-riota 7347  df-ov 7393  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-oi 9451

This theorem is referenced by:  hsmexlem2  10377