Theorem slotfn 17248

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 17246	. . 3 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁))
4	2, 3	eqtri 2786	. 2 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁))
5	1, 4	fnmpti 6678	1 ⊢ 𝐸 Fn V

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3455   ↦ cmpt 5192   Fn wfn 6531  ‘cfv 6536  Slot cslot 17245

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  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 17246

This theorem is used by:  basfn  17277  bj-isrvec  37966