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Theorem dnnumch3lem 39524
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 2818 . 2 (𝑥𝐴 (𝐹 “ {𝑥})) = (𝑥𝐴 (𝐹 “ {𝑥}))
2 sneq 4567 . . . 4 (𝑥 = 𝑤 → {𝑥} = {𝑤})
32imaeq2d 5922 . . 3 (𝑥 = 𝑤 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
43inteqd 4872 . 2 (𝑥 = 𝑤 (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
5 simpr 485 . 2 ((𝜑𝑤𝐴) → 𝑤𝐴)
6 cnvimass 5942 . . . 4 (𝐹 “ {𝑤}) ⊆ dom 𝐹
7 dnnumch.f . . . . . 6 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
87tfr1 8022 . . . . 5 𝐹 Fn On
9 fndm 6448 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
108, 9ax-mp 5 . . . 4 dom 𝐹 = On
116, 10sseqtri 4000 . . 3 (𝐹 “ {𝑤}) ⊆ On
12 dnnumch.a . . . . . 6 (𝜑𝐴𝑉)
13 dnnumch.g . . . . . 6 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
147, 12, 13dnnumch2 39523 . . . . 5 (𝜑𝐴 ⊆ ran 𝐹)
1514sselda 3964 . . . 4 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
16 inisegn0 5954 . . . 4 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
1715, 16sylib 219 . . 3 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
18 oninton 7504 . . 3 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ On)
1911, 17, 18sylancr 587 . 2 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ On)
201, 4, 5, 19fvmptd3 6783 1 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   cint 4867  cmpt 5137  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  Oncon0 6184   Fn wfn 6343  cfv 6348  recscrecs 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-wrecs 7936  df-recs 7997
This theorem is referenced by:  dnnumch3  39525  dnwech  39526
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