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Mirrors > Home > ILE Home > Th. List > f1veqaeq | 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 5771 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑))) | |
2 | fveq2 5517 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝐹‘𝑐) = (𝐹‘𝐶)) | |
3 | 2 | eqeq1d 2186 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝑑))) |
4 | eqeq1 2184 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐 = 𝑑 ↔ 𝐶 = 𝑑)) | |
5 | 3, 4 | imbi12d 234 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑))) |
6 | fveq2 5517 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝐹‘𝑑) = (𝐹‘𝐷)) | |
7 | 6 | eqeq2d 2189 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝐹‘𝐶) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝐷))) |
8 | eqeq2 2187 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝐶 = 𝑑 ↔ 𝐶 = 𝐷)) | |
9 | 7, 8 | imbi12d 234 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
10 | 5, 9 | rspc2v 2856 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
11 | 10 | com12 30 | . . . 4 ⊢ (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
12 | 11 | adantl 277 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑)) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
13 | 1, 12 | sylbi 121 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))) |
14 | 13 | imp 124 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⟶wf 5214 –1-1→wf1 5215 ‘cfv 5218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fv 5226 |
This theorem is referenced by: f1fveq 5775 f1ocnvfvrneq 5785 f1o2ndf1 6231 fidceq 6871 difinfsnlem 7100 difinfsn 7101 iseqf1olemab 10491 iseqf1olemnanb 10492 pwle2 14833 |
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