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Theorem dnnumch3lem 38136
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 simpr 479 . 2 ((𝜑𝑤𝐴) → 𝑤𝐴)
2 cnvimass 5643 . . . 4 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3 dnnumch.f . . . . . 6 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
43tfr1 7663 . . . . 5 𝐹 Fn On
5 fndm 6151 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
64, 5ax-mp 5 . . . 4 dom 𝐹 = On
72, 6sseqtri 3778 . . 3 (𝐹 “ {𝑤}) ⊆ On
8 dnnumch.a . . . . . 6 (𝜑𝐴𝑉)
9 dnnumch.g . . . . . 6 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
103, 8, 9dnnumch2 38135 . . . . 5 (𝜑𝐴 ⊆ ran 𝐹)
1110sselda 3744 . . . 4 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
12 inisegn0 5655 . . . 4 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
1311, 12sylib 208 . . 3 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
14 oninton 7166 . . 3 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ On)
157, 13, 14sylancr 698 . 2 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ On)
16 sneq 4331 . . . . 5 (𝑥 = 𝑤 → {𝑥} = {𝑤})
1716imaeq2d 5624 . . . 4 (𝑥 = 𝑤 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
1817inteqd 4632 . . 3 (𝑥 = 𝑤 (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
19 eqid 2760 . . 3 (𝑥𝐴 (𝐹 “ {𝑥})) = (𝑥𝐴 (𝐹 “ {𝑥}))
2018, 19fvmptg 6443 . 2 ((𝑤𝐴 (𝐹 “ {𝑤}) ∈ On) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
211, 15, 20syl2anc 696 1 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  Vcvv 3340  cdif 3712  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cint 4627  cmpt 4881  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  Oncon0 5884   Fn wfn 6044  cfv 6049  recscrecs 7637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7577  df-recs 7638
This theorem is referenced by:  dnnumch3  38137  dnwech  38138
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