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Mirrors > Home > ILE Home > Th. List > fneq2 | GIF version |
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
fneq2 | ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2187 | . . 3 ⊢ (𝐴 = 𝐵 → (dom 𝐹 = 𝐴 ↔ dom 𝐹 = 𝐵)) | |
2 | 1 | anbi2d 464 | . 2 ⊢ (𝐴 = 𝐵 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))) |
3 | df-fn 5219 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
4 | df-fn 5219 | . 2 ⊢ (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵)) | |
5 | 2, 3, 4 | 3bitr4g 223 | 1 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 dom cdm 4626 Fun wfun 5210 Fn wfn 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-fn 5219 |
This theorem is referenced by: fneq2d 5307 fneq2i 5311 feq2 5349 foeq2 5435 f1o00 5496 eqfnfv2 5614 tfr0dm 6322 tfrlemisucaccv 6325 tfrlemi1 6332 tfrlemi14d 6333 tfrexlem 6334 tfr1onlemsucfn 6340 tfr1onlemsucaccv 6341 tfr1onlembxssdm 6343 tfr1onlembfn 6344 tfr1onlemaccex 6348 tfr1onlemres 6349 ixpeq1 6708 0fz1 10044 |
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