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Theorem fconst2g 5452
 Description: A constant function expressed as a cross product. (Contributed by set.mm contributors, 27-Nov-2007.)
Assertion
Ref Expression
fconst2g (B C → (F:A–→{B} ↔ F = (A × {B})))

Proof of Theorem fconst2g
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 fvconst 5440 . . . . . . 7 ((F:A–→{B} x A) → (Fx) = B)
21adantlr 695 . . . . . 6 (((F:A–→{B} B C) x A) → (Fx) = B)
3 fvconst2g 5451 . . . . . . 7 ((B C x A) → ((A × {B}) ‘x) = B)
43adantll 694 . . . . . 6 (((F:A–→{B} B C) x A) → ((A × {B}) ‘x) = B)
52, 4eqtr4d 2388 . . . . 5 (((F:A–→{B} B C) x A) → (Fx) = ((A × {B}) ‘x))
65ralrimiva 2697 . . . 4 ((F:A–→{B} B C) → x A (Fx) = ((A × {B}) ‘x))
7 ffn 5223 . . . . 5 (F:A–→{B} → F Fn A)
8 fnconstg 5252 . . . . 5 (B C → (A × {B}) Fn A)
9 eqfnfv 5392 . . . . 5 ((F Fn A (A × {B}) Fn A) → (F = (A × {B}) ↔ x A (Fx) = ((A × {B}) ‘x)))
107, 8, 9syl2an 463 . . . 4 ((F:A–→{B} B C) → (F = (A × {B}) ↔ x A (Fx) = ((A × {B}) ‘x)))
116, 10mpbird 223 . . 3 ((F:A–→{B} B C) → F = (A × {B}))
1211expcom 424 . 2 (B C → (F:A–→{B} → F = (A × {B})))
13 fconstg 5251 . . 3 (B C → (A × {B}):A–→{B})
14 feq1 5210 . . 3 (F = (A × {B}) → (F:A–→{B} ↔ (A × {B}):A–→{B}))
1513, 14syl5ibrcom 213 . 2 (B C → (F = (A × {B}) → F:A–→{B}))
1612, 15impbid 183 1 (B C → (F:A–→{B} ↔ F = (A × {B})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {csn 3737   × cxp 4770   Fn wfn 4776  –→wf 4777   ‘cfv 4781 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  df-fv 4795 This theorem is referenced by:  fconst2  5454  fconst5  5455
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