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Theorem fsn2g 6913
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 4527 . . 3 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21feq2d 6491 . 2 (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵𝐹:{𝐴}⟶𝐵))
3 fveq2 6677 . . . 4 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
43eleq1d 2818 . . 3 (𝑎 = 𝐴 → ((𝐹𝑎) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))
5 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
65, 3opeq12d 4770 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
76sneqd 4529 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝐴, (𝐹𝐴)⟩})
87eqeq2d 2750 . . 3 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
94, 8anbi12d 634 . 2 (𝑎 = 𝐴 → (((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
10 vex 3403 . . 3 𝑎 ∈ V
1110fsn2 6911 . 2 (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
122, 9, 11vtoclbg 3473 1 (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  {csn 4517  cop 4523  wf 6336  cfv 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348
This theorem is referenced by:  fsnex  7053  pt1hmeo  22560  k0004val0  41333  difmapsn  42313  fsetsniunop  44105
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