Theorem tailval 36850

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 ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))

Proof of Theorem tailval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tailfval.1	. . . . 5 ⊢ 𝑋 = dom 𝐷
2	1	tailfval 36849	. . . 4 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
3	2	fveq1d 6883	. . 3 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
4	3	adantr 485	. 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
5		id 23	. . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋)
6		imaexg 7909	. . 3 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V)
7		sneq 4598	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
8	7	imaeq2d 6062	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴}))
9		eqid 2761	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))
10	8, 9	fvmptg 6987	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
11	5, 6, 10	syl2anr 608	. 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
12	4, 11	eqtrd 2796	1 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453  {csn 4588   ↦ cmpt 5191  dom cdm 5661   “ cima 5664  ‘cfv 6536  DirRelcdir 18649  tailctail 18650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-dir 18651  df-tail 18652

This theorem is referenced by:  eltail  36851