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| Mirrors > Home > MPE Home > Th. List > feq12i | Structured version Visualization version GIF version | ||
| Description: Equality inference for functions. (Contributed by AV, 7-Feb-2021.) | 
| Ref | Expression | 
|---|---|
| feq12i.1 | ⊢ 𝐹 = 𝐺 | 
| feq12i.2 | ⊢ 𝐴 = 𝐵 | 
| Ref | Expression | 
|---|---|
| feq12i | ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | feq12i.1 | . 2 ⊢ 𝐹 = 𝐺 | |
| 2 | feq12i.2 | . 2 ⊢ 𝐴 = 𝐵 | |
| 3 | eqid 2736 | . 2 ⊢ 𝐶 = 𝐶 | |
| 4 | feq123 6725 | . 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) | |
| 5 | 1, 2, 3, 4 | mp3an 1462 | 1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: climlimsupcex 45789 | 
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