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Theorem fsn2 7137
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 4665 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelcdm 7084 . . . . 5 ((𝐹:{𝐴}⟢𝐡 ∧ 𝐴 ∈ {𝐴}) β†’ (πΉβ€˜π΄) ∈ 𝐡)
42, 3mpan2 687 . . . 4 (𝐹:{𝐴}⟢𝐡 β†’ (πΉβ€˜π΄) ∈ 𝐡)
5 ffn 6718 . . . . 5 (𝐹:{𝐴}⟢𝐡 β†’ 𝐹 Fn {𝐴})
6 dffn3 6731 . . . . . . 7 (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟢ran 𝐹)
76biimpi 215 . . . . . 6 (𝐹 Fn {𝐴} β†’ 𝐹:{𝐴}⟢ran 𝐹)
8 imadmrn 6070 . . . . . . . . 9 (𝐹 β€œ dom 𝐹) = ran 𝐹
9 fndm 6653 . . . . . . . . . 10 (𝐹 Fn {𝐴} β†’ dom 𝐹 = {𝐴})
109imaeq2d 6060 . . . . . . . . 9 (𝐹 Fn {𝐴} β†’ (𝐹 β€œ dom 𝐹) = (𝐹 β€œ {𝐴}))
118, 10eqtr3id 2784 . . . . . . . 8 (𝐹 Fn {𝐴} β†’ ran 𝐹 = (𝐹 β€œ {𝐴}))
12 fnsnfv 6971 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) β†’ {(πΉβ€˜π΄)} = (𝐹 β€œ {𝐴}))
132, 12mpan2 687 . . . . . . . 8 (𝐹 Fn {𝐴} β†’ {(πΉβ€˜π΄)} = (𝐹 β€œ {𝐴}))
1411, 13eqtr4d 2773 . . . . . . 7 (𝐹 Fn {𝐴} β†’ ran 𝐹 = {(πΉβ€˜π΄)})
1514feq3d 6705 . . . . . 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 510 . . 3 (𝐹:{𝐴}⟢𝐡 β†’ ((πΉβ€˜π΄) ∈ 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}))
19 snssi 4812 . . . 4 ((πΉβ€˜π΄) ∈ 𝐡 β†’ {(πΉβ€˜π΄)} βŠ† 𝐡)
20 fss 6735 . . . . 5 ((𝐹:{𝐴}⟢{(πΉβ€˜π΄)} ∧ {(πΉβ€˜π΄)} βŠ† 𝐡) β†’ 𝐹:{𝐴}⟢𝐡)
2120ancoms 457 . . . 4 (({(πΉβ€˜π΄)} βŠ† 𝐡 ∧ 𝐹:{𝐴}⟢{(πΉβ€˜π΄)}) β†’ 𝐹:{𝐴}⟢𝐡)
2219, 21sylan 578 . . 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 6905 . . . 4 (πΉβ€˜π΄) ∈ V
251, 24fsn 7136 . . 3 (𝐹:{𝐴}⟢{(πΉβ€˜π΄)} ↔ 𝐹 = {⟨𝐴, (πΉβ€˜π΄)⟩})
2625anbi2i 621 . 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3949  {csn 4629  βŸ¨cop 4635  dom cdm 5677  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  fsn2g  7139  fnressn  7159  fressnfv  7161  mapsnconst  8890  elixpsn  8935  en1  9025  en1OLD  9026  mat1dimelbas  22195  0spth  29644  wlkl0  29885  ldepsnlinclem1  47275  ldepsnlinclem2  47276  0aryfvalel  47409  1arymaptf1  47417
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