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Theorem fsn2 7136
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
Hypothesis
Ref Expression
fsn2.1 𝐴 ∈ V
Assertion
Ref Expression
fsn2 (𝐹:{𝐴}⟢𝐡 ↔ ((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹 = {⟨𝐴, (πΉβ€˜π΄)⟩}))

Proof of Theorem fsn2
StepHypRef Expression
1 fsn2.1 . . . . . 6 𝐴 ∈ V
21snid 4664 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelcdm 7083 . . . . 5 ((𝐹:{𝐴}⟢𝐡 ∧ 𝐴 ∈ {𝐴}) β†’ (πΉβ€˜π΄) ∈ 𝐡)
42, 3mpan2 689 . . . 4 (𝐹:{𝐴}⟢𝐡 β†’ (πΉβ€˜π΄) ∈ 𝐡)
5 ffn 6717 . . . . 5 (𝐹:{𝐴}⟢𝐡 β†’ 𝐹 Fn {𝐴})
6 dffn3 6730 . . . . . . 7 (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟢ran 𝐹)
76biimpi 215 . . . . . 6 (𝐹 Fn {𝐴} β†’ 𝐹:{𝐴}⟢ran 𝐹)
8 imadmrn 6069 . . . . . . . . 9 (𝐹 β€œ dom 𝐹) = ran 𝐹
9 fndm 6652 . . . . . . . . . 10 (𝐹 Fn {𝐴} β†’ dom 𝐹 = {𝐴})
109imaeq2d 6059 . . . . . . . . 9 (𝐹 Fn {𝐴} β†’ (𝐹 β€œ dom 𝐹) = (𝐹 β€œ {𝐴}))
118, 10eqtr3id 2786 . . . . . . . 8 (𝐹 Fn {𝐴} β†’ ran 𝐹 = (𝐹 β€œ {𝐴}))
12 fnsnfv 6970 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) β†’ {(πΉβ€˜π΄)} = (𝐹 β€œ {𝐴}))
132, 12mpan2 689 . . . . . . . 8 (𝐹 Fn {𝐴} β†’ {(πΉβ€˜π΄)} = (𝐹 β€œ {𝐴}))
1411, 13eqtr4d 2775 . . . . . . 7 (𝐹 Fn {𝐴} β†’ ran 𝐹 = {(πΉβ€˜π΄)})
1514feq3d 6704 . . . . . 6 (𝐹 Fn {𝐴} β†’ (𝐹:{𝐴}⟢ran 𝐹 ↔ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}))
167, 15mpbid 231 . . . . 5 (𝐹 Fn {𝐴} β†’ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)})
175, 16syl 17 . . . 4 (𝐹:{𝐴}⟢𝐡 β†’ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)})
184, 17jca 512 . . 3 (𝐹:{𝐴}⟢𝐡 β†’ ((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}))
19 snssi 4811 . . . 4 ((πΉβ€˜π΄) ∈ 𝐡 β†’ {(πΉβ€˜π΄)} βŠ† 𝐡)
20 fss 6734 . . . . 5 ((𝐹:{𝐴}⟢{(πΉβ€˜π΄)} ∧ {(πΉβ€˜π΄)} βŠ† 𝐡) β†’ 𝐹:{𝐴}⟢𝐡)
2120ancoms 459 . . . 4 (({(πΉβ€˜π΄)} βŠ† 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}) β†’ 𝐹:{𝐴}⟢𝐡)
2219, 21sylan 580 . . 3 (((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}) β†’ 𝐹:{𝐴}⟢𝐡)
2318, 22impbii 208 . 2 (𝐹:{𝐴}⟢𝐡 ↔ ((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}))
24 fvex 6904 . . . 4 (πΉβ€˜π΄) ∈ V
251, 24fsn 7135 . . 3 (𝐹:{𝐴}⟢{(πΉβ€˜π΄)} ↔ 𝐹 = {⟨𝐴, (πΉβ€˜π΄)⟩})
2625anbi2i 623 . 2 (((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}) ↔ ((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹 = {⟨𝐴, (πΉβ€˜π΄)⟩}))
2723, 26bitri 274 1 (𝐹:{𝐴}⟢𝐡 ↔ ((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹 = {⟨𝐴, (πΉβ€˜π΄)⟩}))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  {csn 4628  βŸ¨cop 4634  dom cdm 5676  ran crn 5677   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  fsn2g  7138  fnressn  7158  fressnfv  7160  mapsnconst  8888  elixpsn  8933  en1  9023  en1OLD  9024  mat1dimelbas  21980  0spth  29417  wlkl0  29658  ldepsnlinclem1  47270  ldepsnlinclem2  47271  0aryfvalel  47404  1arymaptf1  47412
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