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Theorem dnnumch3lem 36634
 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 476 . 2 ((𝜑𝑤𝐴) → 𝑤𝐴)
2 cnvimass 5404 . . . 4 (𝐹 “ {𝑤}) ⊆ dom 𝐹
3 dnnumch.f . . . . . 6 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
43tfr1 7380 . . . . 5 𝐹 Fn On
5 fndm 5904 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
64, 5ax-mp 5 . . . 4 dom 𝐹 = On
72, 6sseqtri 3600 . . 3 (𝐹 “ {𝑤}) ⊆ On
8 dnnumch.a . . . . . 6 (𝜑𝐴𝑉)
9 dnnumch.g . . . . . 6 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
103, 8, 9dnnumch2 36633 . . . . 5 (𝜑𝐴 ⊆ ran 𝐹)
1110sselda 3568 . . . 4 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐹)
12 inisegn0 5416 . . . 4 (𝑤 ∈ ran 𝐹 ↔ (𝐹 “ {𝑤}) ≠ ∅)
1311, 12sylib 207 . . 3 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ≠ ∅)
14 oninton 6892 . . 3 (((𝐹 “ {𝑤}) ⊆ On ∧ (𝐹 “ {𝑤}) ≠ ∅) → (𝐹 “ {𝑤}) ∈ On)
157, 13, 14sylancr 694 . 2 ((𝜑𝑤𝐴) → (𝐹 “ {𝑤}) ∈ On)
16 sneq 4135 . . . . 5 (𝑥 = 𝑤 → {𝑥} = {𝑤})
1716imaeq2d 5385 . . . 4 (𝑥 = 𝑤 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
1817inteqd 4415 . . 3 (𝑥 = 𝑤 (𝐹 “ {𝑥}) = (𝐹 “ {𝑤}))
19 eqid 2610 . . 3 (𝑥𝐴 (𝐹 “ {𝑥})) = (𝑥𝐴 (𝐹 “ {𝑥}))
2018, 19fvmptg 6189 . 2 ((𝑤𝐴 (𝐹 “ {𝑤}) ∈ On) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
211, 15, 20syl2anc 691 1 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∩ cint 4410   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Oncon0 5640   Fn wfn 5799  ‘cfv 5804  recscrecs 7354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355 This theorem is referenced by:  dnnumch3  36635  dnwech  36636
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