Theorem bj-evalidval 33031

Description: Closed general form of strndxid 15885. Both sides are equal to (𝐹‘𝐴) by bj-evalid 33028 and bj-evalval 33027 respectively, but bj-evalidval 33031 adds something to bj-evalid 33028 and bj-evalval 33027 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)

Assertion

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

Proof of Theorem bj-evalidval

Step	Hyp	Ref	Expression
1		bj-evalid 33028	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
2	1	fveq2d 6195	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹‘𝐴))
3	2	3adant3 1081	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹‘𝐴))
4		bj-evalval 33027	. . . 4 ⊢ (𝐹 ∈ 𝑈 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))
5	4	eqcomd 2628	. . 3 ⊢ (𝐹 ∈ 𝑈 → (𝐹‘𝐴) = (Slot 𝐴‘𝐹))
6	5	3ad2ant3 1084	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑈) → (𝐹‘𝐴) = (Slot 𝐴‘𝐹))
7	3, 6	eqtrd 2656	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴‘𝐹))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   I cid 5023   ↾ cres 5116  ‘cfv 5888  Slot cslot 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by: (None)