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Mirrors > Home > MPE Home > Th. List > feu | Structured version Visualization version GIF version |
Description: There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
feu | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6517 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fneu2 6465 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) | |
3 | 1, 2 | sylan 582 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) |
4 | opelf 6542 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
5 | 4 | simprd 498 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → 𝑦 ∈ 𝐵) |
6 | 5 | ex 415 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 565 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
8 | 7 | eubidv 2671 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
10 | 3, 9 | mpbid 234 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) |
11 | df-reu 3148 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) | |
12 | 10, 11 | sylibr 236 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ∃!weu 2652 ∃!wreu 3143 〈cop 4576 Fn wfn 6353 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: fsn 6900 f1ofveu 7154 |
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