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Theorem tailval 32063
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 32062 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
32fveq1d 6160 . . 3 (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
43adantr 481 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
5 id 22 . . 3 (𝐴𝑋𝐴𝑋)
6 imaexg 7065 . . 3 (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V)
7 sneq 4165 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
87imaeq2d 5435 . . . 4 (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴}))
9 eqid 2621 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
108, 9fvmptg 6247 . . 3 ((𝐴𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
115, 6, 10syl2anr 495 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
124, 11eqtrd 2655 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155  cmpt 4683  dom cdm 5084  cima 5087  cfv 5857  DirRelcdir 17168  tailctail 17169
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 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-dir 17170  df-tail 17171
This theorem is referenced by:  eltail  32064
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