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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 2180 | . . 3 ⊢ (𝐴 = 𝐵 → (dom 𝐹 = 𝐴 ↔ dom 𝐹 = 𝐵)) | |
2 | 1 | anbi2d 461 | . 2 ⊢ (𝐴 = 𝐵 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))) |
3 | df-fn 5201 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
4 | df-fn 5201 | . 2 ⊢ (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵)) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 dom cdm 4611 Fun wfun 5192 Fn wfn 5193 |
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-5 1440 ax-gen 1442 ax-4 1503 ax-17 1519 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 df-fn 5201 |
This theorem is referenced by: fneq2d 5289 fneq2i 5293 feq2 5331 foeq2 5417 f1o00 5477 eqfnfv2 5594 tfr0dm 6301 tfrlemisucaccv 6304 tfrlemi1 6311 tfrlemi14d 6312 tfrexlem 6313 tfr1onlemsucfn 6319 tfr1onlemsucaccv 6320 tfr1onlembxssdm 6322 tfr1onlembfn 6323 tfr1onlemaccex 6327 tfr1onlemres 6328 ixpeq1 6687 0fz1 10001 |
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