Theorem slotfn 17243

Description: A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypothesis

Ref	Expression
strfvnd.c	⊢ 𝐸 = Slot 𝑁

Assertion

Ref	Expression
slotfn	⊢ 𝐸 Fn V

Proof of Theorem slotfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6894	. 2 ⊢ (𝑥‘𝑁) ∈ V
2		strfvnd.c	. . 3 ⊢ 𝐸 = Slot 𝑁
3		df-slot 17241	. . 3 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁))
4	2, 3	eqtri 2784	. 2 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁))
5	1, 4	fnmpti 6678	1 ⊢ 𝐸 Fn V

Syntax hints:   = wceq 1568  Vcvv 3453   ↦ cmpt 5191   Fn wfn 6531  ‘cfv 6536  Slot cslot 17240

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-sep 5256  ax-nul 5268  ax-pr 5404

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-rab 3415  df-v 3455  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-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-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-slot 17241

This theorem is referenced by:  basfn  17272  bj-isrvec  37904