Theorem oif 9495

Description: The order isomorphism of the well-order 𝑅 on 𝐴 is a function. (Contributed by Mario Carneiro, 23-May-2015.)

Hypothesis

Ref	Expression
oicl.1	⊢ 𝐹 = OrdIso(𝑅, 𝐴)

Assertion

Ref	Expression
oif	⊢ 𝐹:dom 𝐹⟶𝐴

Proof of Theorem oif

Dummy variables 𝑢 𝑡 𝑣 𝑥 ℎ 𝑗 𝑤 𝑧 𝑓 𝑖 𝑟 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) = recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
2		eqid 2765	. . . . 5 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		eqid 2765	. . . . 5 ⊢ (ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)) = (ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
4	1, 2, 3	ordtypecbv 9482	. . . 4 ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
5		eqid 2765	. . . 4 ⊢ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) “ 𝑥)𝑧𝑅𝑡} = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) “ 𝑥)𝑧𝑅𝑡}
6		oicl.1	. . . 4 ⊢ 𝐹 = OrdIso(𝑅, 𝐴)
7		simpl 488	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 We 𝐴)
8		simpr 490	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
9	4, 2, 3, 5, 6, 7, 8	ordtypelem5 9487	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (Ord dom 𝐹 ∧ 𝐹:dom 𝐹⟶𝐴))
10	9	simprd 501	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹:dom 𝐹⟶𝐴)
11		f0 6763	. . 3 ⊢ ∅:∅⟶𝐴
12	6	oi0 9493	. . . 4 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅)
13	12	dmeqd 5897	. . . . 5 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14		dm0 5912	. . . . 5 ⊢ dom ∅ = ∅
15	13, 14	eqtrdi 2816	. . . 4 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = ∅)
16	12, 15	feq12d 6697	. . 3 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹:dom 𝐹⟶𝐴 ↔ ∅:∅⟶𝐴))
17	11, 16	mpbiri 261	. 2 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹:dom 𝐹⟶𝐴)
18	10, 17	pm2.61i 184	1 ⊢ 𝐹:dom 𝐹⟶𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457  ∅c0 4286   class class class wbr 5111   ↦ cmpt 5194   Se wse 5614   We wwe 5615  dom cdm 5663  ran crn 5664   “ cima 5666  Ord word 6363  Oncon0 6364  ⟶wf 6536  ℩crio 7372  recscrecs 8359  OrdIsocoi 9474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-oi 9475

This theorem is used by:  oismo  9505  cantnfle  9643  cantnflt  9644  cantnfres  9649  cantnfp1lem3  9652  cantnflem1b  9658  cantnflem1  9661  wemapwe  9669  cnfcomlem  9671  cnfcom  9672  cnfcom3lem  9675  cnfcom3  9676  hsmexlem1  10421  hsmexlem2  10422  fpwwe2lem7  10633