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Theorem fconst5 5455
 Description: Two ways to express that a function is constant. (Contributed by set.mm contributors, 27-Nov-2007.)
Assertion
Ref Expression
fconst5 ((F Fn A A) → (F = (A × {B}) ↔ ran F = {B}))

Proof of Theorem fconst5
StepHypRef Expression
1 rneq 4956 . . . 4 (F = (A × {B}) → ran F = ran (A × {B}))
2 rnxp 5051 . . . . 5 (A → ran (A × {B}) = {B})
32eqeq2d 2364 . . . 4 (A → (ran F = ran (A × {B}) ↔ ran F = {B}))
41, 3syl5ib 210 . . 3 (A → (F = (A × {B}) → ran F = {B}))
54adantl 452 . 2 ((F Fn A A) → (F = (A × {B}) → ran F = {B}))
6 df-fo 4793 . . . . . . 7 (F:Aonto→{B} ↔ (F Fn A ran F = {B}))
7 fof 5269 . . . . . . 7 (F:Aonto→{B} → F:A–→{B})
86, 7sylbir 204 . . . . . 6 ((F Fn A ran F = {B}) → F:A–→{B})
9 fconst2g 5452 . . . . . 6 (B V → (F:A–→{B} ↔ F = (A × {B})))
108, 9syl5ib 210 . . . . 5 (B V → ((F Fn A ran F = {B}) → F = (A × {B})))
1110exp3a 425 . . . 4 (B V → (F Fn A → (ran F = {B} → F = (A × {B}))))
1211adantrd 454 . . 3 (B V → ((F Fn A A) → (ran F = {B} → F = (A × {B}))))
13 rneq0 4970 . . . . . . 7 (F = ↔ ran F = )
1413a1i 10 . . . . . 6 B V → (F = ↔ ran F = ))
15 snprc 3788 . . . . . . . . . 10 B V ↔ {B} = )
1615biimpi 186 . . . . . . . . 9 B V → {B} = )
1716xpeq2d 4808 . . . . . . . 8 B V → (A × {B}) = (A × ))
18 xp0 5044 . . . . . . . 8 (A × ) =
1917, 18syl6eq 2401 . . . . . . 7 B V → (A × {B}) = )
2019eqeq2d 2364 . . . . . 6 B V → (F = (A × {B}) ↔ F = ))
2116eqeq2d 2364 . . . . . 6 B V → (ran F = {B} ↔ ran F = ))
2214, 20, 213bitr4d 276 . . . . 5 B V → (F = (A × {B}) ↔ ran F = {B}))
2322biimprd 214 . . . 4 B V → (ran F = {B} → F = (A × {B})))
2423a1d 22 . . 3 B V → ((F Fn A A) → (ran F = {B} → F = (A × {B}))))
2512, 24pm2.61i 156 . 2 ((F Fn A A) → (ran F = {B} → F = (A × {B})))
265, 25impbid 183 1 ((F Fn A A) → (F = (A × {B}) ↔ ran F = {B}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859  ∅c0 3550  {csn 3737   × cxp 4770  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779 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-fo 4793  df-fv 4795 This theorem is referenced by: (None)
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