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Theorem f1o00 5317
 Description: One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.)
Assertion
Ref Expression
f1o00 (F:1-1-ontoA ↔ (F = A = ))

Proof of Theorem f1o00
StepHypRef Expression
1 dff1o4 5294 . 2 (F:1-1-ontoA ↔ (F Fn F Fn A))
2 fn0 5202 . . . . . 6 (F Fn F = )
32biimpi 186 . . . . 5 (F Fn F = )
43adantr 451 . . . 4 ((F Fn F Fn A) → F = )
5 dm0 4918 . . . . 5 dom =
6 cnveq 4886 . . . . . . . . . 10 (F = F = )
7 cnv0 5031 . . . . . . . . . 10 =
86, 7syl6eq 2401 . . . . . . . . 9 (F = F = )
92, 8sylbi 187 . . . . . . . 8 (F Fn F = )
109fneq1d 5175 . . . . . . 7 (F Fn → (F Fn A Fn A))
1110biimpa 470 . . . . . 6 ((F Fn F Fn A) → Fn A)
12 fndm 5182 . . . . . 6 ( Fn A → dom = A)
1311, 12syl 15 . . . . 5 ((F Fn F Fn A) → dom = A)
145, 13syl5reqr 2400 . . . 4 ((F Fn F Fn A) → A = )
154, 14jca 518 . . 3 ((F Fn F Fn A) → (F = A = ))
162biimpri 197 . . . . 5 (F = F Fn )
1716adantr 451 . . . 4 ((F = A = ) → F Fn )
18 eqid 2353 . . . . . 6 =
19 fn0 5202 . . . . . 6 ( Fn = )
2018, 19mpbir 200 . . . . 5 Fn
218fneq1d 5175 . . . . . 6 (F = → (F Fn A Fn A))
22 fneq2 5174 . . . . . 6 (A = → ( Fn A Fn ))
2321, 22sylan9bb 680 . . . . 5 ((F = A = ) → (F Fn A Fn ))
2420, 23mpbiri 224 . . . 4 ((F = A = ) → F Fn A)
2517, 24jca 518 . . 3 ((F = A = ) → (F Fn F Fn A))
2615, 25impbii 180 . 2 ((F Fn F Fn A) ↔ (F = A = ))
271, 26bitri 240 1 (F:1-1-ontoA ↔ (F = A = ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ∅c0 3550  ◡ccnv 4771  dom cdm 4772   Fn wfn 4776  –1-1-onto→wf1o 4780 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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  fo00  5318  en0  6042
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