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Theorem tailval 34207
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 34206 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
32fveq1d 6678 . . 3 (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
43adantr 484 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
5 id 22 . . 3 (𝐴𝑋𝐴𝑋)
6 imaexg 7648 . . 3 (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V)
7 sneq 4526 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
87imaeq2d 5903 . . . 4 (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴}))
9 eqid 2738 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
108, 9fvmptg 6775 . . 3 ((𝐴𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
115, 6, 10syl2anr 600 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
124, 11eqtrd 2773 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3398  {csn 4516  cmpt 5110  dom cdm 5525  cima 5528  cfv 6339  DirRelcdir 17956  tailctail 17957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-dir 17958  df-tail 17959
This theorem is referenced by:  eltail  34208
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