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Theorem tailval 32695
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 32694 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
32fveq1d 6355 . . 3 (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
43adantr 472 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
5 id 22 . . 3 (𝐴𝑋𝐴𝑋)
6 imaexg 7269 . . 3 (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V)
7 sneq 4331 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
87imaeq2d 5624 . . . 4 (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴}))
9 eqid 2760 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
108, 9fvmptg 6443 . . 3 ((𝐴𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
115, 6, 10syl2anr 496 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
124, 11eqtrd 2794 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cmpt 4881  dom cdm 5266  cima 5269  cfv 6049  DirRelcdir 17449  tailctail 17450
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-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  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-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-dir 17451  df-tail 17452
This theorem is referenced by:  eltail  32696
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