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Theorem dnnumch3 38112
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 5695 . . . . 5 (𝐹 “ {𝑥}) ⊆ dom 𝐹
2 dnnumch.f . . . . . . 7 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
32tfr1 7725 . . . . . 6 𝐹 Fn On
4 fndm 6197 . . . . . 6 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 5 . . . . 5 dom 𝐹 = On
61, 5sseqtri 3834 . . . 4 (𝐹 “ {𝑥}) ⊆ On
7 dnnumch.a . . . . . . 7 (𝜑𝐴𝑉)
8 dnnumch.g . . . . . . 7 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
92, 7, 8dnnumch2 38110 . . . . . 6 (𝜑𝐴 ⊆ ran 𝐹)
109sselda 3798 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ ran 𝐹)
11 inisegn0 5707 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ (𝐹 “ {𝑥}) ≠ ∅)
1210, 11sylib 209 . . . 4 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ≠ ∅)
13 oninton 7226 . . . 4 (((𝐹 “ {𝑥}) ⊆ On ∧ (𝐹 “ {𝑥}) ≠ ∅) → (𝐹 “ {𝑥}) ∈ On)
146, 12, 13sylancr 577 . . 3 ((𝜑𝑥𝐴) → (𝐹 “ {𝑥}) ∈ On)
1514fmpttd 6603 . 2 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
162, 7, 8dnnumch3lem 38111 . . . . . 6 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1716adantrr 699 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
182, 7, 8dnnumch3lem 38111 . . . . . 6 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
1918adantrl 698 . . . . 5 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2017, 19eqeq12d 2821 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})))
21 fveq2 6404 . . . . . . 7 ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
2221adantl 469 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = (𝐹 (𝐹 “ {𝑤})))
23 cnvimass 5695 . . . . . . . . . . 11 (𝐹 “ {𝑣}) ⊆ dom 𝐹
2423, 5sseqtri 3834 . . . . . . . . . 10 (𝐹 “ {𝑣}) ⊆ On
259sselda 3798 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ ran 𝐹)
26 inisegn0 5707 . . . . . . . . . . 11 (𝑣 ∈ ran 𝐹 ↔ (𝐹 “ {𝑣}) ≠ ∅)
2725, 26sylib 209 . . . . . . . . . 10 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ≠ ∅)
28 onint 7221 . . . . . . . . . 10 (((𝐹 “ {𝑣}) ⊆ On ∧ (𝐹 “ {𝑣}) ≠ ∅) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
2924, 27, 28sylancr 577 . . . . . . . . 9 ((𝜑𝑣𝐴) → (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}))
30 fniniseg 6556 . . . . . . . . . . 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 486 . . . . . . . . 9 ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑣}) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3329, 32syl 17 . . . . . . . 8 ((𝜑𝑣𝐴) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3433adantrr 699 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
3534adantr 468 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑣})) = 𝑣)
36 cnvimass 5695 . . . . . . . . . . 11 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3736, 5sseqtri 3834 . . . . . . . . . 10 (𝐹 “ {𝑤}) ⊆ On
389sselda 3798 . . . . . . . . . . 11 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
39 inisegn0 5707 . . . . . . . . . . 11 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
4038, 39sylib 209 . . . . . . . . . 10 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
41 onint 7221 . . . . . . . . . 10 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
4237, 40, 41sylancr 577 . . . . . . . . 9 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}))
43 fniniseg 6556 . . . . . . . . . . 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 486 . . . . . . . . 9 ( (𝐹 “ {𝑤}) ∈ (𝐹 “ {𝑤}) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4642, 45syl 17 . . . . . . . 8 ((𝜑𝑤𝐴) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4746adantrl 698 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4847adantr 468 . . . . . 6 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → (𝐹 (𝐹 “ {𝑤})) = 𝑤)
4922, 35, 483eqtr3d 2848 . . . . 5 (((𝜑 ∧ (𝑣𝐴𝑤𝐴)) ∧ (𝐹 “ {𝑣}) = (𝐹 “ {𝑤})) → 𝑣 = 𝑤)
5049ex 399 . . . 4 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) = (𝐹 “ {𝑤}) → 𝑣 = 𝑤))
5120, 50sylbid 231 . . 3 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
5251ralrimivva 3159 . 2 (𝜑 → ∀𝑣𝐴𝑤𝐴 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
53 dff13 6732 . 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 574 1 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wne 2978  wral 3096  Vcvv 3391  cdif 3766  wss 3769  c0 4116  𝒫 cpw 4351  {csn 4370   cint 4669  cmpt 4923  ccnv 5310  dom cdm 5311  ran crn 5312  cima 5314  Oncon0 5936   Fn wfn 6092  wf 6093  1-1wf1 6094  cfv 6097  recscrecs 7699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-wrecs 7638  df-recs 7700
This theorem is referenced by:  dnwech  38113
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