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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 33012 | . . 3 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) |
5 | 4 | expdimp 455 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 → 𝐵 = 𝐶)) |
6 | breq2 5069 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝐹𝐵 ↔ 𝐴𝐹𝐶)) | |
7 | 6 | biimpcd 251 | . . 3 ⊢ (𝐴𝐹𝐵 → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
8 | 7 | adantl 484 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
9 | 5, 8 | impbid 214 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5065 Fun wfun 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-cnv 5562 df-co 5563 df-fun 6356 |
This theorem is referenced by: (None) |
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