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Mirrors > Home > MPE Home > Th. List > f1ocnvfvrneq | Structured version Visualization version GIF version |
Description: If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
Ref | Expression |
---|---|
f1ocnvfvrneq | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f1orn 6229 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
2 | f1ocnv 6230 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
3 | f1of1 6217 | . . 3 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹:ran 𝐹–1-1→𝐴) | |
4 | f1veqaeq 6597 | . . . 4 ⊢ ((◡𝐹:ran 𝐹–1-1→𝐴 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)) | |
5 | 4 | ex 449 | . . 3 ⊢ (◡𝐹:ran 𝐹–1-1→𝐴 → ((𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))) |
6 | 1, 2, 3, 5 | 4syl 19 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))) |
7 | 6 | imp 444 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1564 ∈ wcel 2071 ◡ccnv 5185 ran crn 5187 –1-1→wf1 5966 –1-1-onto→wf1o 5968 ‘cfv 5969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pr 4979 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ral 2987 df-rex 2988 df-rab 2991 df-v 3274 df-sbc 3510 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-nul 3992 df-if 4163 df-sn 4254 df-pr 4256 df-op 4260 df-uni 4513 df-br 4729 df-opab 4789 df-id 5096 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 |
This theorem is referenced by: (None) |
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