![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . 2 ⊢ 𝐶 = 𝐶 | |
4 | feq123 6685 | . 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1461 | 1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ⟶wf 6519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3426 df-v 3468 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-sn 4614 df-pr 4616 df-op 4620 df-br 5133 df-opab 5195 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-fun 6525 df-fn 6526 df-f 6527 |
This theorem is referenced by: climlimsupcex 44170 |
Copyright terms: Public domain | W3C validator |