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Theorem dnnumch3lem 37093
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 477 . 2 ((𝜑𝑤𝐴) → 𝑤𝐴)
2 cnvimass 5444 . . . 4 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3 dnnumch.f . . . . . 6 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
43tfr1 7438 . . . . 5 𝐹 Fn On
5 fndm 5948 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
64, 5ax-mp 5 . . . 4 dom 𝐹 = On
72, 6sseqtri 3616 . . 3 (𝐹 “ {𝑤}) ⊆ On
8 dnnumch.a . . . . . 6 (𝜑𝐴𝑉)
9 dnnumch.g . . . . . 6 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
103, 8, 9dnnumch2 37092 . . . . 5 (𝜑𝐴 ⊆ ran 𝐹)
1110sselda 3583 . . . 4 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
12 inisegn0 5456 . . . 4 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
1311, 12sylib 208 . . 3 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
14 oninton 6947 . . 3 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ On)
157, 13, 14sylancr 694 . 2 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ On)
16 sneq 4158 . . . . 5 (𝑥 = 𝑤 → {𝑥} = {𝑤})
1716imaeq2d 5425 . . . 4 (𝑥 = 𝑤 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
1817inteqd 4445 . . 3 (𝑥 = 𝑤 (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
19 eqid 2621 . . 3 (𝑥𝐴 (𝐹 “ {𝑥})) = (𝑥𝐴 (𝐹 “ {𝑥}))
2018, 19fvmptg 6237 . 2 ((𝑤𝐴 (𝐹 “ {𝑤}) ∈ On) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
211, 15, 20syl2anc 692 1 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cint 4440  cmpt 4673  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  Oncon0 5682   Fn wfn 5842  cfv 5847  recscrecs 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-wrecs 7352  df-recs 7413
This theorem is referenced by:  dnnumch3  37094  dnwech  37095
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