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Theorem dnnumch3 42745
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 6083 . . . . 5 (𝐹 “ {𝑥}) ⊆ dom 𝐹
2 dnnumch.f . . . . . . 7 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
32tfr1 8419 . . . . . 6 𝐹 Fn On
43fndmi 6656 . . . . 5 dom 𝐹 = On
51, 4sseqtri 4015 . . . 4 (𝐹 “ {𝑥}) ⊆ On
6 dnnumch.a . . . . . . 7 (𝜑𝐴𝑉)
7 dnnumch.g . . . . . . 7 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
82, 6, 7dnnumch2 42743 . . . . . 6 (𝜑𝐴 ⊆ ran 𝐹)
98sselda 3978 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ ran 𝐹)
10 inisegn0 6100 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ (𝐹 “ {𝑥}) ≠ ∅)
119, 10sylib 217 . . . 4 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ≠ ∅)
12 oninton 7796 . . . 4 (((𝐹 “ {𝑥}) ⊆ On ∧ (𝐹 “ {𝑥}) ≠ ∅) → (𝐹 “ {𝑥}) ∈ On)
135, 11, 12sylancr 585 . . 3 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ∈ On)
1413fmpttd 7121 . 2 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
152, 6, 7dnnumch3lem 42744 . . . . . 6 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1615adantrr 715 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
172, 6, 7dnnumch3lem 42744 . . . . . 6 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
1817adantrl 714 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
1916, 18eqeq12d 2742 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})))
20 fveq2 6893 . . . . . . 7 ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
2120adantl 480 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
22 cnvimass 6083 . . . . . . . . . . 11 (𝐹 “ {𝑣}) ⊆ dom 𝐹
2322, 4sseqtri 4015 . . . . . . . . . 10 (𝐹 “ {𝑣}) ⊆ On
248sselda 3978 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ ran 𝐹)
25 inisegn0 6100 . . . . . . . . . . 11 (𝑣 ∈ ran 𝐹 ↔ (𝐹 “ {𝑣}) ≠ ∅)
2624, 25sylib 217 . . . . . . . . . 10 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ≠ ∅)
27 onint 7791 . . . . . . . . . 10 (((𝐹 “ {𝑣}) ⊆ On ∧ (𝐹 “ {𝑣}) ≠ ∅) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
2823, 26, 27sylancr 585 . . . . . . . . 9 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
29 fniniseg 7065 . . . . . . . . . . 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 495 . . . . . . . . 9 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3228, 31syl 17 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3332adantrr 715 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3433adantr 479 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
35 cnvimass 6083 . . . . . . . . . . 11 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3635, 4sseqtri 4015 . . . . . . . . . 10 (𝐹 “ {𝑤}) ⊆ On
378sselda 3978 . . . . . . . . . . 11 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
38 inisegn0 6100 . . . . . . . . . . 11 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
3937, 38sylib 217 . . . . . . . . . 10 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
40 onint 7791 . . . . . . . . . 10 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
4136, 39, 40sylancr 585 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
42 fniniseg 7065 . . . . . . . . . . 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 495 . . . . . . . . 9 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4541, 44syl 17 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4645adantrl 714 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4746adantr 479 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4821, 34, 473eqtr3d 2774 . . . . 5 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → 𝑣 = 𝑤)
4948ex 411 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → 𝑣 = 𝑤))
5019, 49sylbid 239 . . 3 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
5150ralrimivva 3191 . 2 (𝜑 → ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
52 dff13 7262 . 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 581 1 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  Vcvv 3462  cdif 3943  wss 3946  c0 4322  𝒫 cpw 4597  {csn 4623   cint 4946  cmpt 5228  ccnv 5673  dom cdm 5674  ran crn 5675  cima 5677  Oncon0 6368   Fn wfn 6541  wf 6542  1-1wf1 6543  cfv 6546  recscrecs 8392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393
This theorem is referenced by:  dnwech  42746
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