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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 7012 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑))) | |
2 | fveqeq2 6678 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝑑))) | |
3 | eqeq1 2825 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 = 𝑑 ↔ 𝐶 = 𝑑)) | |
4 | 2, 3 | imbi12d 347 | . . . . 5 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑))) |
5 | fveq2 6669 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝐹‘𝑑) = (𝐹‘𝐷)) | |
6 | 5 | eqeq2d 2832 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝐹‘𝐶) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝐷))) |
7 | eqeq2 2833 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 = 𝑑 ↔ 𝐶 = 𝐷)) | |
8 | 6, 7 | imbi12d 347 | . . . . 5 ⊢ (𝑑 = 𝐷 → (((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
9 | 4, 8 | rspc2v 3632 | . . . 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 507 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
12 | 11 | imp 409 | 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 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6350 –1-1→wf1 6351 ‘cfv 6354 |
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-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 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-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fv 6362 |
This theorem is referenced by: f1cofveqaeq 7015 f1cofveqaeqALT 7016 2f1fvneq 7017 f1fveq 7019 f1prex 7039 f1ocnvfvrneq 7041 f1o2ndf1 7817 symgfvne 18508 f1ghm0to0 19491 f1rhm0to0OLD 19492 mat2pmatf1 21336 f1otrg 26656 uspgr2wlkeq 27426 pthdivtx 27509 spthdep 27514 spthonepeq 27532 usgr2trlncl 27540 ccatf1 30625 swrdf1 30630 cycpmrn 30785 f1resveqaeq 32358 poimirlem1 34892 poimirlem9 34900 poimirlem22 34913 mblfinlem2 34929 isomuspgrlem1 43991 |
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