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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 6688 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹) | |
2 | fnfun 6669 | . . . . 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 4035 | . . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {〈𝐴, (𝐹‘𝐴)〉}) |
6 | 5 | sseld 3994 | . 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {〈𝐴, (𝐹‘𝐴)〉})) |
7 | elsni 4648 | . 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 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 〈cop 4637 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: fnsnb 7186 fnsnbt 42250 |
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