![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6660 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹) | |
2 | fnfun 6640 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹) | |
3 | funressn 7150 | . . . . 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 4014 | . . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹‘𝐴)⟩}) |
6 | 5 | sseld 3974 | . 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩})) |
7 | elsni 4638 | . 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 1533 ∈ wcel 2098 ⊆ wss 3941 {csn 4621 ⟨cop 4627 ↾ cres 5669 Fun wfun 6528 Fn wfn 6529 ‘cfv 6534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 |
This theorem is referenced by: fnsnb 7157 fnsnbt 41586 |
Copyright terms: Public domain | W3C validator |