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Mirrors > Home > MPE Home > Th. List > sloteq | Structured version Visualization version GIF version |
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 6897 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵)) | |
2 | 1 | mpteq2dv 5250 | . 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵))) |
3 | df-slot 17151 | . 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | df-slot 17151 | . 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵)) | |
5 | 2, 3, 4 | 3eqtr4g 2793 | 1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 Vcvv 3471 ↦ cmpt 5231 ‘cfv 6548 Slot cslot 17150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6500 df-fv 6556 df-slot 17151 |
This theorem is referenced by: ndxid 17166 |
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