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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 6846 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵)) | |
2 | 1 | mpteq2dv 5211 | . 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵))) |
3 | df-slot 17062 | . 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | df-slot 17062 | . 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵)) | |
5 | 2, 3, 4 | 3eqtr4g 2798 | 1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3447 ↦ cmpt 5192 ‘cfv 6500 Slot cslot 17061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-iota 6452 df-fv 6508 df-slot 17062 |
This theorem is referenced by: ndxid 17077 |
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