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| Mirrors > Home > MPE Home > Th. List > Mathboxes > funbreq | Structured version Visualization version GIF version | ||
| Description: An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.) | 
| Ref | Expression | 
|---|---|
| funbreq.1 | ⊢ 𝐴 ∈ V | 
| funbreq.2 | ⊢ 𝐵 ∈ V | 
| funbreq.3 | ⊢ 𝐶 ∈ V | 
| Ref | Expression | 
|---|---|
| funbreq | ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funbreq.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 2 | funbreq.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 3 | funbreq.3 | . . . 4 ⊢ 𝐶 ∈ V | |
| 4 | 1, 2, 3 | fununiq 35769 | . . 3 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) | 
| 5 | 4 | expdimp 452 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 → 𝐵 = 𝐶)) | 
| 6 | breq2 5147 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝐹𝐵 ↔ 𝐴𝐹𝐶)) | |
| 7 | 6 | biimpcd 249 | . . 3 ⊢ (𝐴𝐹𝐵 → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) | 
| 8 | 7 | adantl 481 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) | 
| 9 | 5, 8 | impbid 212 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 Fun wfun 6555 | 
| 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-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 | 
| This theorem is referenced by: (None) | 
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