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| Mirrors > Home > MPE Home > Th. List > feq2 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) |
| Ref | Expression |
|---|---|
| feq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fneq2 6660 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | |
| 2 | 1 | anbi1d 631 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))) |
| 3 | df-f 6565 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
| 4 | df-f 6565 | . 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) | |
| 5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ⊆ wss 3951 ran crn 5686 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-fn 6564 df-f 6565 |
| This theorem is referenced by: feq23 6719 feq2d 6722 feq2i 6728 f00 6790 f0dom0 6792 f1eq2 6800 fressnfv 7180 poseq 8183 soseq 8184 mapvalg 8876 fsetexb 8904 map0g 8924 ac6sfi 9320 cofsmo 10309 axcc4dom 10481 ac6sg 10528 isghm 19233 isghmOLD 19234 pjdm2 21731 cmpcovf 23399 ulmval 26423 elno2 27699 noreson 27705 measval 34199 isrnmeas 34201 bj-finsumval0 37286 mbfresfi 37673 sn-isghm 42683 dfno2 43441 relpeq4 44968 stoweidlem62 46077 hoidmvval0b 46605 vonioo 46697 vonicc 46700 f102g 48761 f1mo 48762 |
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