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Theorem dff3 5420
 Description: Alternate definition of a mapping. (Contributed by set.mm contributors, 20-Mar-2007.)
Assertion
Ref Expression
dff3 (F:A–→B ↔ (F (A × B) x A ∃!y xFy))
Distinct variable groups:   x,y,A   x,B,y   x,F,y

Proof of Theorem dff3
StepHypRef Expression
1 fssxp 5232 . . 3 (F:A–→BF (A × B))
2 fdm 5226 . . . . . . . 8 (F:A–→B → dom F = A)
32eleq2d 2420 . . . . . . 7 (F:A–→B → (x dom Fx A))
43biimpar 471 . . . . . 6 ((F:A–→B x A) → x dom F)
5 eldm 4898 . . . . . 6 (x dom Fy xFy)
64, 5sylib 188 . . . . 5 ((F:A–→B x A) → y xFy)
7 ffun 5225 . . . . . . 7 (F:A–→B → Fun F)
87adantr 451 . . . . . 6 ((F:A–→B x A) → Fun F)
9 funmo 5125 . . . . . 6 (Fun F∃*y xFy)
108, 9syl 15 . . . . 5 ((F:A–→B x A) → ∃*y xFy)
11 eu5 2242 . . . . 5 (∃!y xFy ↔ (y xFy ∃*y xFy))
126, 10, 11sylanbrc 645 . . . 4 ((F:A–→B x A) → ∃!y xFy)
1312ralrimiva 2697 . . 3 (F:A–→Bx A ∃!y xFy)
141, 13jca 518 . 2 (F:A–→B → (F (A × B) x A ∃!y xFy))
15 df-ral 2619 . . . . . . 7 (x A ∃!y xFyx(x A∃!y xFy))
16 dmss 4906 . . . . . . . . . . . . . . 15 (F (A × B) → dom F dom (A × B))
17 dmxpss 5052 . . . . . . . . . . . . . . 15 dom (A × B) A
1816, 17syl6ss 3284 . . . . . . . . . . . . . 14 (F (A × B) → dom F A)
1918sseld 3272 . . . . . . . . . . . . 13 (F (A × B) → (x dom Fx A))
205, 19syl5bir 209 . . . . . . . . . . . 12 (F (A × B) → (y xFyx A))
2120con3d 125 . . . . . . . . . . 11 (F (A × B) → (¬ x A → ¬ y xFy))
22 pm2.21 100 . . . . . . . . . . . 12 y xFy → (y xFy∃!y xFy))
23 df-mo 2209 . . . . . . . . . . . 12 (∃*y xFy ↔ (y xFy∃!y xFy))
2422, 23sylibr 203 . . . . . . . . . . 11 y xFy∃*y xFy)
2521, 24syl6 29 . . . . . . . . . 10 (F (A × B) → (¬ x A∃*y xFy))
2625a1dd 42 . . . . . . . . 9 (F (A × B) → (¬ x A → ((x A∃!y xFy) → ∃*y xFy)))
27 pm2.27 35 . . . . . . . . . 10 (x A → ((x A∃!y xFy) → ∃!y xFy))
28 eumo 2244 . . . . . . . . . 10 (∃!y xFy∃*y xFy)
2927, 28syl6 29 . . . . . . . . 9 (x A → ((x A∃!y xFy) → ∃*y xFy))
3026, 29pm2.61d2 152 . . . . . . . 8 (F (A × B) → ((x A∃!y xFy) → ∃*y xFy))
3130alimdv 1621 . . . . . . 7 (F (A × B) → (x(x A∃!y xFy) → x∃*y xFy))
3215, 31syl5bi 208 . . . . . 6 (F (A × B) → (x A ∃!y xFyx∃*y xFy))
3332imp 418 . . . . 5 ((F (A × B) x A ∃!y xFy) → x∃*y xFy)
34 dffun6 5124 . . . . 5 (Fun Fx∃*y xFy)
3533, 34sylibr 203 . . . 4 ((F (A × B) x A ∃!y xFy) → Fun F)
3618adantr 451 . . . . 5 ((F (A × B) x A ∃!y xFy) → dom F A)
37 euex 2227 . . . . . . . . 9 (∃!y xFyy xFy)
3837, 5sylibr 203 . . . . . . . 8 (∃!y xFyx dom F)
3938ralimi 2689 . . . . . . 7 (x A ∃!y xFyx A x dom F)
40 dfss3 3263 . . . . . . 7 (A dom Fx A x dom F)
4139, 40sylibr 203 . . . . . 6 (x A ∃!y xFyA dom F)
4241adantl 452 . . . . 5 ((F (A × B) x A ∃!y xFy) → A dom F)
4336, 42eqssd 3289 . . . 4 ((F (A × B) x A ∃!y xFy) → dom F = A)
44 df-fn 4790 . . . 4 (F Fn A ↔ (Fun F dom F = A))
4535, 43, 44sylanbrc 645 . . 3 ((F (A × B) x A ∃!y xFy) → F Fn A)
46 rnss 4959 . . . . 5 (F (A × B) → ran F ran (A × B))
47 rnxpss 5053 . . . . 5 ran (A × B) B
4846, 47syl6ss 3284 . . . 4 (F (A × B) → ran F B)
4948adantr 451 . . 3 ((F (A × B) x A ∃!y xFy) → ran F B)
50 df-f 4791 . . 3 (F:A–→B ↔ (F Fn A ran F B))
5145, 49, 50sylanbrc 645 . 2 ((F (A × B) x A ∃!y xFy) → F:A–→B)
5214, 51impbii 180 1 (F:A–→B ↔ (F (A × B) x A ∃!y xFy))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2614   ⊆ wss 3257   class class class wbr 4639   × cxp 4770  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-fun 4789  df-fn 4790  df-f 4791 This theorem is referenced by:  dff4  5421
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