![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f1veqaeq | Structured version Visualization version GIF version |
Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
Ref | Expression |
---|---|
f1veqaeq | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff13 7262 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑))) | |
2 | fveqeq2 6902 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝑑))) | |
3 | eqeq1 2730 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 = 𝑑 ↔ 𝐶 = 𝑑)) | |
4 | 2, 3 | imbi12d 343 | . . . . 5 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑))) |
5 | fveq2 6893 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝐹‘𝑑) = (𝐹‘𝐷)) | |
6 | 5 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝐹‘𝐶) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝐷))) |
7 | eqeq2 2738 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 = 𝑑 ↔ 𝐶 = 𝐷)) | |
8 | 6, 7 | imbi12d 343 | . . . . 5 ⊢ (𝑑 = 𝐷 → (((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
9 | 4, 8 | rspc2v 3618 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
10 | 9 | com12 32 | . . 3 ⊢ (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
11 | 1, 10 | simplbiim 503 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
12 | 11 | imp 405 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⟶wf 6542 –1-1→wf1 6543 ‘cfv 6546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fv 6554 |
This theorem is referenced by: f1cofveqaeq 7265 f1cofveqaeqALT 7266 2f1fvneq 7267 f1fveq 7269 f1cdmsn 7288 f1prex 7290 f1ocnvfvrneq 7292 f1o2ndf1 8128 f1ghm0to0 19235 symgfvne 19374 mat2pmatf1 22719 f1otrg 28795 uspgr2wlkeq 29580 pthdivtx 29663 spthdep 29668 spthonepeq 29686 usgr2trlncl 29694 ccatf1 32815 swrdf1 32823 cycpmrn 33025 f1resveqaeq 34935 poimirlem1 37335 poimirlem9 37343 poimirlem22 37356 mblfinlem2 37372 ricdrng1 42218 isuspgrim0lem 47486 |
Copyright terms: Public domain | W3C validator |