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Theorem tailval 36356
Description: The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Hypothesis
Ref Expression
tailfval.1 𝑋 = dom 𝐷
Assertion
Ref Expression
tailval ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))

Proof of Theorem tailval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tailfval.1 . . . . 5 𝑋 = dom 𝐷
21tailfval 36355 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
32fveq1d 6909 . . 3 (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
43adantr 480 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
5 id 22 . . 3 (𝐴𝑋𝐴𝑋)
6 imaexg 7936 . . 3 (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V)
7 sneq 4641 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
87imaeq2d 6080 . . . 4 (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴}))
9 eqid 2735 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
108, 9fvmptg 7014 . . 3 ((𝐴𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
115, 6, 10syl2anr 597 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
124, 11eqtrd 2775 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cmpt 5231  dom cdm 5689  cima 5692  cfv 6563  DirRelcdir 18652  tailctail 18653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-dir 18654  df-tail 18655
This theorem is referenced by:  eltail  36357
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