MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsn2g Structured version   Visualization version   GIF version

Theorem fsn2g 6633
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
Assertion
Ref Expression
fsn2g (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))

Proof of Theorem fsn2g
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 sneq 4379 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21feq2d 6243 . . 3 (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵𝐹:{𝐴}⟶𝐵))
3 fveq2 6412 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
43eleq1d 2864 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))
5 eqidd 2801 . . . . 5 (𝑎 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . . . 7 (𝑎 = 𝐴𝑎 = 𝐴)
76, 3opeq12d 4602 . . . . . 6 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
87sneqd 4381 . . . . 5 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝐴, (𝐹𝐴)⟩})
95, 8eqeq12d 2815 . . . 4 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
104, 9anbi12d 625 . . 3 (𝑎 = 𝐴 → (((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
112, 10bibi12d 337 . 2 (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩})) ↔ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))))
12 vex 3389 . . 3 𝑎 ∈ V
1312fsn2 6631 . 2 (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
1411, 13vtoclg 3454 1 (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  {csn 4369  cop 4375  wf 6098  cfv 6102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110
This theorem is referenced by:  fsnex  6767  pt1hmeo  21937  k0004val0  39229  difmapsn  40151
  Copyright terms: Public domain W3C validator