Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnnumch3 Structured version   Visualization version   GIF version

Theorem dnnumch3 39509
Description: Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a (𝜑𝐴𝑉)
dnnumch.g (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
Assertion
Ref Expression
dnnumch3 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem dnnumch3
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 5946 . . . . 5 (𝐹 “ {𝑥}) ⊆ dom 𝐹
2 dnnumch.f . . . . . . 7 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
32tfr1 8027 . . . . . 6 𝐹 Fn On
4 fndm 6451 . . . . . 6 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 5 . . . . 5 dom 𝐹 = On
61, 5sseqtri 4006 . . . 4 (𝐹 “ {𝑥}) ⊆ On
7 dnnumch.a . . . . . . 7 (𝜑𝐴𝑉)
8 dnnumch.g . . . . . . 7 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
92, 7, 8dnnumch2 39507 . . . . . 6 (𝜑𝐴 ⊆ ran 𝐹)
109sselda 3970 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ ran 𝐹)
11 inisegn0 5958 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ (𝐹 “ {𝑥}) ≠ ∅)
1210, 11sylib 219 . . . 4 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ≠ ∅)
13 oninton 7506 . . . 4 (((𝐹 “ {𝑥}) ⊆ On ∧ (𝐹 “ {𝑥}) ≠ ∅) → (𝐹 “ {𝑥}) ∈ On)
146, 12, 13sylancr 587 . . 3 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ∈ On)
1514fmpttd 6874 . 2 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
162, 7, 8dnnumch3lem 39508 . . . . . 6 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1716adantrr 713 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
182, 7, 8dnnumch3lem 39508 . . . . . 6 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
1918adantrl 712 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2017, 19eqeq12d 2840 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})))
21 fveq2 6666 . . . . . . 7 ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
2221adantl 482 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
23 cnvimass 5946 . . . . . . . . . . 11 (𝐹 “ {𝑣}) ⊆ dom 𝐹
2423, 5sseqtri 4006 . . . . . . . . . 10 (𝐹 “ {𝑣}) ⊆ On
259sselda 3970 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ ran 𝐹)
26 inisegn0 5958 . . . . . . . . . . 11 (𝑣 ∈ ran 𝐹 ↔ (𝐹 “ {𝑣}) ≠ ∅)
2725, 26sylib 219 . . . . . . . . . 10 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ≠ ∅)
28 onint 7501 . . . . . . . . . 10 (((𝐹 “ {𝑣}) ⊆ On ∧ (𝐹 “ {𝑣}) ≠ ∅) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
2924, 27, 28sylancr 587 . . . . . . . . 9 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
30 fniniseg 6825 . . . . . . . . . . 11 (𝐹 Fn On → ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) ↔ ( (𝐹 “ {𝑣}) ∈ On ∧ (𝐹 (𝐹 “ {𝑣})) = 𝑣)))
313, 30ax-mp 5 . . . . . . . . . 10 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) ↔ ( (𝐹 “ {𝑣}) ∈ On ∧ (𝐹 (𝐹 “ {𝑣})) = 𝑣))
3231simprbi 497 . . . . . . . . 9 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3329, 32syl 17 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3433adantrr 713 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3534adantr 481 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
36 cnvimass 5946 . . . . . . . . . . 11 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3736, 5sseqtri 4006 . . . . . . . . . 10 (𝐹 “ {𝑤}) ⊆ On
389sselda 3970 . . . . . . . . . . 11 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
39 inisegn0 5958 . . . . . . . . . . 11 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
4038, 39sylib 219 . . . . . . . . . 10 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
41 onint 7501 . . . . . . . . . 10 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
4237, 40, 41sylancr 587 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
43 fniniseg 6825 . . . . . . . . . . 11 (𝐹 Fn On → ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) ↔ ( (𝐹 “ {𝑤}) ∈ On ∧ (𝐹 (𝐹 “ {𝑤})) = 𝑤)))
443, 43ax-mp 5 . . . . . . . . . 10 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) ↔ ( (𝐹 “ {𝑤}) ∈ On ∧ (𝐹 (𝐹 “ {𝑤})) = 𝑤))
4544simprbi 497 . . . . . . . . 9 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4642, 45syl 17 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4746adantrl 712 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4847adantr 481 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4922, 35, 483eqtr3d 2868 . . . . 5 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → 𝑣 = 𝑤)
5049ex 413 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → 𝑣 = 𝑤))
5120, 50sylbid 241 . . 3 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
5251ralrimivva 3195 . 2 (𝜑 → ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
53 dff13 7010 . 2 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On ↔ ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On ∧ ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤)))
5415, 52, 53sylanbrc 583 1 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  wne 3020  wral 3142  Vcvv 3499  cdif 3936  wss 3939  c0 4294  𝒫 cpw 4541  {csn 4563   cint 4873  cmpt 5142  ccnv 5552  dom cdm 5553  ran crn 5554  cima 5556  Oncon0 6188   Fn wfn 6346  wf 6347  1-1wf1 6348  cfv 6351  recscrecs 8001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-wrecs 7941  df-recs 8002
This theorem is referenced by:  dnwech  39510
  Copyright terms: Public domain W3C validator