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Theorem fsn2 5434
 Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by set.mm contributors, 19-May-2004.)
Hypothesis
Ref Expression
fsn2.1 A V
Assertion
Ref Expression
fsn2 (F:{A}–→B ↔ ((FA) B F = {A, (FA)}))

Proof of Theorem fsn2
StepHypRef Expression
1 fsn2.1 . . . . . 6 A V
21snid 3760 . . . . 5 A {A}
3 ffvelrn 5415 . . . . 5 ((F:{A}–→B A {A}) → (FA) B)
42, 3mpan2 652 . . . 4 (F:{A}–→B → (FA) B)
5 ffn 5223 . . . . 5 (F:{A}–→BF Fn {A})
6 dffn3 5229 . . . . . . 7 (F Fn {A} ↔ F:{A}–→ran F)
76biimpi 186 . . . . . 6 (F Fn {A} → F:{A}–→ran F)
8 imadmrn 5008 . . . . . . . . 9 (F “ dom F) = ran F
9 fndm 5182 . . . . . . . . . 10 (F Fn {A} → dom F = {A})
109imaeq2d 4942 . . . . . . . . 9 (F Fn {A} → (F “ dom F) = (F “ {A}))
118, 10syl5eqr 2399 . . . . . . . 8 (F Fn {A} → ran F = (F “ {A}))
12 fnsnfv 5373 . . . . . . . . 9 ((F Fn {A} A {A}) → {(FA)} = (F “ {A}))
132, 12mpan2 652 . . . . . . . 8 (F Fn {A} → {(FA)} = (F “ {A}))
1411, 13eqtr4d 2388 . . . . . . 7 (F Fn {A} → ran F = {(FA)})
15 feq3 5212 . . . . . . 7 (ran F = {(FA)} → (F:{A}–→ran FF:{A}–→{(FA)}))
1614, 15syl 15 . . . . . 6 (F Fn {A} → (F:{A}–→ran FF:{A}–→{(FA)}))
177, 16mpbid 201 . . . . 5 (F Fn {A} → F:{A}–→{(FA)})
185, 17syl 15 . . . 4 (F:{A}–→BF:{A}–→{(FA)})
194, 18jca 518 . . 3 (F:{A}–→B → ((FA) B F:{A}–→{(FA)}))
20 snssi 3852 . . . 4 ((FA) B → {(FA)} B)
21 fss 5230 . . . . 5 ((F:{A}–→{(FA)} {(FA)} B) → F:{A}–→B)
2221ancoms 439 . . . 4 (({(FA)} B F:{A}–→{(FA)}) → F:{A}–→B)
2320, 22sylan 457 . . 3 (((FA) B F:{A}–→{(FA)}) → F:{A}–→B)
2419, 23impbii 180 . 2 (F:{A}–→B ↔ ((FA) B F:{A}–→{(FA)}))
25 fvex 5339 . . . 4 (FA) V
261, 25fsn 5432 . . 3 (F:{A}–→{(FA)} ↔ F = {A, (FA)})
2726anbi2i 675 . 2 (((FA) B F:{A}–→{(FA)}) ↔ ((FA) B F = {A, (FA)}))
2824, 27bitri 240 1 (F:{A}–→B ↔ ((FA) B F = {A, (FA)}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟨cop 4561   “ cima 4722  dom cdm 4772  ran crn 4773   Fn wfn 4776  –→wf 4777   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795 This theorem is referenced by:  fnressn  5438  fressnfv  5439  1cnc  6139
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