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Mirrors > Home > ILE Home > Th. List > sloteq | 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 5421 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵)) | |
2 | 1 | mpteq2dv 4019 | . 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵))) |
3 | df-slot 11963 | . 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | df-slot 11963 | . 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵)) | |
5 | 2, 3, 4 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2686 ↦ cmpt 3989 ‘cfv 5123 Slot cslot 11958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-iota 5088 df-fv 5131 df-slot 11963 |
This theorem is referenced by: ndxid 11983 |
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