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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 2823 | . 2 ⊢ 𝐶 = 𝐶 | |
4 | feq123 6506 | . 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1457 | 1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: climlimsupcex 42057 |
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