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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 32909 | . . 3 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) |
5 | 4 | expdimp 453 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 → 𝐵 = 𝐶)) |
6 | breq2 5061 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝐹𝐵 ↔ 𝐴𝐹𝐶)) | |
7 | 6 | biimpcd 250 | . . 3 ⊢ (𝐴𝐹𝐵 → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
8 | 7 | adantl 482 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
9 | 5, 8 | impbid 213 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: (None) |
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