Theorem tailval 36602

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 36601	. . . 4 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
3	2	fveq1d 6836	. . 3 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
4	3	adantr 481	. 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴))
5		id 22	. . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋)
6		imaexg 7860	. . 3 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V)
7		sneq 4572	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
8	7	imaeq2d 6019	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴}))
9		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))
10	8, 9	fvmptg 6940	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
11	5, 6, 10	syl2anr 603	. 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴}))
12	4, 11	eqtrd 2775	1 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562   ↦ cmpt 5160  dom cdm 5625   “ cima 5628  ‘cfv 6492  DirRelcdir 18558  tailctail 18559

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-dir 18560  df-tail 18561

This theorem is referenced by:  eltail  36603