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Theorem dnnumch3 37934
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 5520 . . . . 5 (𝐹 “ {𝑥}) ⊆ dom 𝐹
2 dnnumch.f . . . . . . 7 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
32tfr1 7538 . . . . . 6 𝐹 Fn On
4 fndm 6028 . . . . . 6 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 5 . . . . 5 dom 𝐹 = On
61, 5sseqtri 3670 . . . 4 (𝐹 “ {𝑥}) ⊆ On
7 dnnumch.a . . . . . . 7 (𝜑𝐴𝑉)
8 dnnumch.g . . . . . . 7 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
92, 7, 8dnnumch2 37932 . . . . . 6 (𝜑𝐴 ⊆ ran 𝐹)
109sselda 3636 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ ran 𝐹)
11 inisegn0 5532 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ (𝐹 “ {𝑥}) ≠ ∅)
1210, 11sylib 208 . . . 4 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ≠ ∅)
13 oninton 7042 . . . 4 (((𝐹 “ {𝑥}) ⊆ On ∧ (𝐹 “ {𝑥}) ≠ ∅) → (𝐹 “ {𝑥}) ∈ On)
146, 12, 13sylancr 696 . . 3 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ∈ On)
15 eqid 2651 . . 3 (𝑥𝐴 (𝐹 “ {𝑥})) = (𝑥𝐴 (𝐹 “ {𝑥}))
1614, 15fmptd 6425 . 2 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
172, 7, 8dnnumch3lem 37933 . . . . . 6 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1817adantrr 753 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
192, 7, 8dnnumch3lem 37933 . . . . . 6 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2019adantrl 752 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2118, 20eqeq12d 2666 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})))
22 fveq2 6229 . . . . . . 7 ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
2322adantl 481 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
24 cnvimass 5520 . . . . . . . . . . 11 (𝐹 “ {𝑣}) ⊆ dom 𝐹
2524, 5sseqtri 3670 . . . . . . . . . 10 (𝐹 “ {𝑣}) ⊆ On
269sselda 3636 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ ran 𝐹)
27 inisegn0 5532 . . . . . . . . . . 11 (𝑣 ∈ ran 𝐹 ↔ (𝐹 “ {𝑣}) ≠ ∅)
2826, 27sylib 208 . . . . . . . . . 10 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ≠ ∅)
29 onint 7037 . . . . . . . . . 10 (((𝐹 “ {𝑣}) ⊆ On ∧ (𝐹 “ {𝑣}) ≠ ∅) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
3025, 28, 29sylancr 696 . . . . . . . . 9 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
31 fniniseg 6378 . . . . . . . . . . 11 (𝐹 Fn On → ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) ↔ ( (𝐹 “ {𝑣}) ∈ On ∧ (𝐹 (𝐹 “ {𝑣})) = 𝑣)))
323, 31ax-mp 5 . . . . . . . . . 10 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) ↔ ( (𝐹 “ {𝑣}) ∈ On ∧ (𝐹 (𝐹 “ {𝑣})) = 𝑣))
3332simprbi 479 . . . . . . . . 9 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3430, 33syl 17 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3534adantrr 753 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3635adantr 480 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
37 cnvimass 5520 . . . . . . . . . . 11 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3837, 5sseqtri 3670 . . . . . . . . . 10 (𝐹 “ {𝑤}) ⊆ On
399sselda 3636 . . . . . . . . . . 11 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
40 inisegn0 5532 . . . . . . . . . . 11 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
4139, 40sylib 208 . . . . . . . . . 10 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
42 onint 7037 . . . . . . . . . 10 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
4338, 41, 42sylancr 696 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
44 fniniseg 6378 . . . . . . . . . . 11 (𝐹 Fn On → ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) ↔ ( (𝐹 “ {𝑤}) ∈ On ∧ (𝐹 (𝐹 “ {𝑤})) = 𝑤)))
453, 44ax-mp 5 . . . . . . . . . 10 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) ↔ ( (𝐹 “ {𝑤}) ∈ On ∧ (𝐹 (𝐹 “ {𝑤})) = 𝑤))
4645simprbi 479 . . . . . . . . 9 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4743, 46syl 17 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4847adantrl 752 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4948adantr 480 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
5023, 36, 493eqtr3d 2693 . . . . 5 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → 𝑣 = 𝑤)
5150ex 449 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → 𝑣 = 𝑤))
5221, 51sylbid 230 . . 3 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
5352ralrimivva 3000 . 2 (𝜑 → ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
54 dff13 6552 . 2 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On ↔ ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On ∧ ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤)))
5516, 53, 54sylanbrc 699 1 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cint 4507  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Oncon0 5761   Fn wfn 5921  wf 5922  1-1wf1 5923  cfv 5926  recscrecs 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-wrecs 7452  df-recs 7513
This theorem is referenced by:  dnwech  37935
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