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| Description: Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| sloteq | ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6906 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵)) | |
| 2 | 1 | mpteq2dv 5244 | . 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵))) | 
| 3 | df-slot 17219 | . 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
| 4 | df-slot 17219 | . 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵)) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 Slot cslot 17218 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-iota 6514 df-fv 6569 df-slot 17219 | 
| This theorem is referenced by: ndxid 17234 | 
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