Theorem bj-evalid 37356

Description: The evaluation at a set of the identity function is that set. General form of ndxarg 17137. The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.)

Assertion

Ref	Expression
bj-evalid	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Proof of Theorem bj-evalid

Step	Hyp	Ref	Expression
1		resiexg 7866	. . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑉) ∈ V)
2		bj-evalval 37355	. . 3 ⊢ (( I ↾ 𝑉) ∈ V → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝑉 ∈ 𝑊 → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴))
4		fvresi 7131	. 2 ⊢ (𝐴 ∈ 𝑉 → (( I ↾ 𝑉)‘𝐴) = 𝐴)
5	3, 4	sylan9eq 2792	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   I cid 5528   ↾ cres 5636  ‘cfv 6502  Slot cslot 17122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6458  df-fun 6504  df-fv 6510  df-slot 17123

This theorem is referenced by:  bj-ndxarg  37357  bj-evalidval  37358