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Theorem dnnumch3lem 39507
 Description: Value of the ordinal injection 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
dnnumch3lem ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
Distinct variable groups:   𝑤,𝐹,𝑥,𝑦   𝑤,𝐺,𝑥,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦,𝑧   𝜑,𝑥,𝑤
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem dnnumch3lem
StepHypRef Expression
1 eqid 2825 . 2 (𝑥𝐴 (𝐹 “ {𝑥})) = (𝑥𝐴 (𝐹 “ {𝑥}))
2 sneq 4573 . . . 4 (𝑥 = 𝑤 → {𝑥} = {𝑤})
32imaeq2d 5926 . . 3 (𝑥 = 𝑤 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
43inteqd 4878 . 2 (𝑥 = 𝑤 (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
5 simpr 485 . 2 ((𝜑𝑤𝐴) → 𝑤𝐴)
6 cnvimass 5946 . . . 4 (𝐹 “ {𝑤}) ⊆ dom 𝐹
7 dnnumch.f . . . . . 6 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
87tfr1 8027 . . . . 5 𝐹 Fn On
9 fndm 6451 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
108, 9ax-mp 5 . . . 4 dom 𝐹 = On
116, 10sseqtri 4006 . . 3 (𝐹 “ {𝑤}) ⊆ On
12 dnnumch.a . . . . . 6 (𝜑𝐴𝑉)
13 dnnumch.g . . . . . 6 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
147, 12, 13dnnumch2 39506 . . . . 5 (𝜑𝐴 ⊆ ran 𝐹)
1514sselda 3970 . . . 4 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
16 inisegn0 5958 . . . 4 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
1715, 16sylib 219 . . 3 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
18 oninton 7506 . . 3 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ On)
1911, 17, 18sylancr 587 . 2 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ On)
201, 4, 5, 19fvmptd3 6786 1 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ≠ 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  ‘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 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  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 2619  df-eu 2651  df-clab 2804  df-cleq 2818  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:  dnnumch3  39508  dnwech  39509
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