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Theorem dnnumch3 43059
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 6100 . . . . 5 (𝐹 “ {𝑥}) ⊆ dom 𝐹
2 dnnumch.f . . . . . . 7 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
32tfr1 8437 . . . . . 6 𝐹 Fn On
43fndmi 6672 . . . . 5 dom 𝐹 = On
51, 4sseqtri 4032 . . . 4 (𝐹 “ {𝑥}) ⊆ On
6 dnnumch.a . . . . . . 7 (𝜑𝐴𝑉)
7 dnnumch.g . . . . . . 7 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
82, 6, 7dnnumch2 43057 . . . . . 6 (𝜑𝐴 ⊆ ran 𝐹)
98sselda 3983 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ ran 𝐹)
10 inisegn0 6116 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ (𝐹 “ {𝑥}) ≠ ∅)
119, 10sylib 218 . . . 4 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ≠ ∅)
12 oninton 7815 . . . 4 (((𝐹 “ {𝑥}) ⊆ On ∧ (𝐹 “ {𝑥}) ≠ ∅) → (𝐹 “ {𝑥}) ∈ On)
135, 11, 12sylancr 587 . . 3 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ∈ On)
1413fmpttd 7135 . 2 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
152, 6, 7dnnumch3lem 43058 . . . . . 6 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1615adantrr 717 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
172, 6, 7dnnumch3lem 43058 . . . . . 6 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
1817adantrl 716 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
1916, 18eqeq12d 2753 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})))
20 fveq2 6906 . . . . . . 7 ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
2120adantl 481 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
22 cnvimass 6100 . . . . . . . . . . 11 (𝐹 “ {𝑣}) ⊆ dom 𝐹
2322, 4sseqtri 4032 . . . . . . . . . 10 (𝐹 “ {𝑣}) ⊆ On
248sselda 3983 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ ran 𝐹)
25 inisegn0 6116 . . . . . . . . . . 11 (𝑣 ∈ ran 𝐹 ↔ (𝐹 “ {𝑣}) ≠ ∅)
2624, 25sylib 218 . . . . . . . . . 10 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ≠ ∅)
27 onint 7810 . . . . . . . . . 10 (((𝐹 “ {𝑣}) ⊆ On ∧ (𝐹 “ {𝑣}) ≠ ∅) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
2823, 26, 27sylancr 587 . . . . . . . . 9 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
29 fniniseg 7080 . . . . . . . . . . 11 (𝐹 Fn On → ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) ↔ ( (𝐹 “ {𝑣}) ∈ On ∧ (𝐹 (𝐹 “ {𝑣})) = 𝑣)))
303, 29ax-mp 5 . . . . . . . . . 10 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) ↔ ( (𝐹 “ {𝑣}) ∈ On ∧ (𝐹 (𝐹 “ {𝑣})) = 𝑣))
3130simprbi 496 . . . . . . . . 9 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3228, 31syl 17 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3332adantrr 717 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3433adantr 480 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
35 cnvimass 6100 . . . . . . . . . . 11 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3635, 4sseqtri 4032 . . . . . . . . . 10 (𝐹 “ {𝑤}) ⊆ On
378sselda 3983 . . . . . . . . . . 11 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
38 inisegn0 6116 . . . . . . . . . . 11 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
3937, 38sylib 218 . . . . . . . . . 10 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
40 onint 7810 . . . . . . . . . 10 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
4136, 39, 40sylancr 587 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
42 fniniseg 7080 . . . . . . . . . . 11 (𝐹 Fn On → ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) ↔ ( (𝐹 “ {𝑤}) ∈ On ∧ (𝐹 (𝐹 “ {𝑤})) = 𝑤)))
433, 42ax-mp 5 . . . . . . . . . 10 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) ↔ ( (𝐹 “ {𝑤}) ∈ On ∧ (𝐹 (𝐹 “ {𝑤})) = 𝑤))
4443simprbi 496 . . . . . . . . 9 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4541, 44syl 17 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4645adantrl 716 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4746adantr 480 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4821, 34, 473eqtr3d 2785 . . . . 5 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → 𝑣 = 𝑤)
4948ex 412 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → 𝑣 = 𝑤))
5019, 49sylbid 240 . . 3 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
5150ralrimivva 3202 . 2 (𝜑 → ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
52 dff13 7275 . 2 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On ↔ ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On ∧ ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤)))
5314, 51, 52sylanbrc 583 1 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cint 4946  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Oncon0 6384   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  recscrecs 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by:  dnwech  43060
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