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Theorem fnsnb 6909
Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)
Hypothesis
Ref Expression
fnsnb.1 𝐴 ∈ V
Assertion
Ref Expression
fnsnb (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})

Proof of Theorem fnsnb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnsnr 6908 . . . . 5 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
2 df-fn 6331 . . . . . . . 8 (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3 fnsnb.1 . . . . . . . . . . 11 𝐴 ∈ V
43snid 4564 . . . . . . . . . 10 𝐴 ∈ {𝐴}
5 eleq2 2881 . . . . . . . . . 10 (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹𝐴 ∈ {𝐴}))
64, 5mpbiri 261 . . . . . . . . 9 (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
76anim2i 619 . . . . . . . 8 ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹𝐴 ∈ dom 𝐹))
82, 7sylbi 220 . . . . . . 7 (𝐹 Fn {𝐴} → (Fun 𝐹𝐴 ∈ dom 𝐹))
9 funfvop 6801 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
108, 9syl 17 . . . . . 6 (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2880 . . . . . 6 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 250 . . . . 5 (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
131, 12impbid 215 . . . 4 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
14 velsn 4544 . . . 4 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14syl6bbr 292 . . 3 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2799 . 2 (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
17 fvex 6662 . . . 4 (𝐹𝐴) ∈ V
183, 17fnsn 6386 . . 3 {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}
19 fneq1 6418 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2018, 19mpbiri 261 . 2 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → 𝐹 Fn {𝐴})
2116, 20impbii 212 1 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  {csn 4528  cop 4534  dom cdm 5523  Fun wfun 6322   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336
This theorem is referenced by:  fnprb  6952
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