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Mirrors > Home > MPE Home > Th. List > fneq2 | Structured version Visualization version 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 2830 | . . 3 ⊢ (𝐴 = 𝐵 → (dom 𝐹 = 𝐴 ↔ dom 𝐹 = 𝐵)) | |
2 | 1 | anbi2d 628 | . 2 ⊢ (𝐴 = 𝐵 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))) |
3 | df-fn 6351 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
4 | df-fn 6351 | . 2 ⊢ (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵)) | |
5 | 2, 3, 4 | 3bitr4g 315 | 1 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 dom cdm 5548 Fun wfun 6342 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-fn 6351 |
This theorem is referenced by: fneq2d 6440 fneq2i 6444 feq2 6489 foeq2 6580 f1o00 6642 eqfnfv2 6795 wfrlem1 7943 wfrlem15 7958 tfrlem12 8014 ixpeq1 8460 ac5 9887 0fz1 12915 esumcvgsum 31246 bnj90 31891 bnj919 31937 bnj535 32061 bnj1463 32224 frrlem1 33020 frrlem13 33032 fnchoice 41163 |
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