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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 35272 | . . 3 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) |
5 | 4 | expdimp 452 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 → 𝐵 = 𝐶)) |
6 | breq2 5145 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝐹𝐵 ↔ 𝐴𝐹𝐶)) | |
7 | 6 | biimpcd 248 | . . 3 ⊢ (𝐴𝐹𝐵 → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
8 | 7 | adantl 481 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
9 | 5, 8 | impbid 211 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 Fun wfun 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-fun 6538 |
This theorem is referenced by: (None) |
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