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| Mirrors > Home > MPE Home > Th. List > fnsnr | Structured version Visualization version GIF version | ||
| Description: If a class belongs to a function on a singleton, then that class is the obvious ordered pair. Note that this theorem also holds when 𝐴 is a proper class, but its meaning is then different. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
| Ref | Expression |
|---|---|
| fnsnr | ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 = 〈𝐴, (𝐹‘𝐴)〉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnresdm 6687 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹) | |
| 2 | fnfun 6668 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹) | |
| 3 | funressn 7179 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {〈𝐴, (𝐹‘𝐴)〉}) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {〈𝐴, (𝐹‘𝐴)〉}) |
| 5 | 1, 4 | eqsstrrd 4019 | . . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {〈𝐴, (𝐹‘𝐴)〉}) |
| 6 | 5 | sseld 3982 | . 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {〈𝐴, (𝐹‘𝐴)〉})) |
| 7 | elsni 4643 | . 2 ⊢ (𝐵 ∈ {〈𝐴, (𝐹‘𝐴)〉} → 𝐵 = 〈𝐴, (𝐹‘𝐴)〉) | |
| 8 | 6, 7 | syl6 35 | 1 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 = 〈𝐴, (𝐹‘𝐴)〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 〈cop 4632 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: fnsnb 7186 fnsnbt 42271 |
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