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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 33125 | . . 3 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) |
5 | 4 | expdimp 456 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 → 𝐵 = 𝐶)) |
6 | breq2 5034 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝐹𝐵 ↔ 𝐴𝐹𝐶)) | |
7 | 6 | biimpcd 252 | . . 3 ⊢ (𝐴𝐹𝐵 → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
8 | 7 | adantl 485 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐵 = 𝐶 → 𝐴𝐹𝐶)) |
9 | 5, 8 | impbid 215 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-cnv 5527 df-co 5528 df-fun 6326 |
This theorem is referenced by: (None) |
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