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Theorem fconst5 7225
Description: Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
Assertion
Ref Expression
fconst5 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))

Proof of Theorem fconst5
StepHypRef Expression
1 rneq 5944 . . . 4 (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = ran (𝐴 × {𝐵}))
2 rnxp 6183 . . . . 5 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
32eqeq2d 2737 . . . 4 (𝐴 ≠ ∅ → (ran 𝐹 = ran (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
41, 3imbitrid 243 . . 3 (𝐴 ≠ ∅ → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
54adantl 480 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
6 df-fo 6562 . . . . . . 7 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
7 fof 6817 . . . . . . 7 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
86, 7sylbir 234 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
9 fconst2g 7222 . . . . . 6 (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
108, 9imbitrid 243 . . . . 5 (𝐵 ∈ V → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹 = (𝐴 × {𝐵})))
1110expd 414 . . . 4 (𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
1211adantrd 490 . . 3 (𝐵 ∈ V → ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
13 fnrel 6664 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
14 snprc 4726 . . . . . 6 𝐵 ∈ V ↔ {𝐵} = ∅)
15 relrn0 5978 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
1615biimprd 247 . . . . . . . . 9 (Rel 𝐹 → (ran 𝐹 = ∅ → 𝐹 = ∅))
1716adantl 480 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = ∅ → 𝐹 = ∅))
18 eqeq2 2738 . . . . . . . . 9 ({𝐵} = ∅ → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
1918adantr 479 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
20 xpeq2 5705 . . . . . . . . . . 11 ({𝐵} = ∅ → (𝐴 × {𝐵}) = (𝐴 × ∅))
21 xp0 6171 . . . . . . . . . . 11 (𝐴 × ∅) = ∅
2220, 21eqtrdi 2782 . . . . . . . . . 10 ({𝐵} = ∅ → (𝐴 × {𝐵}) = ∅)
2322eqeq2d 2737 . . . . . . . . 9 ({𝐵} = ∅ → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2423adantr 479 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2517, 19, 243imtr4d 293 . . . . . . 7 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
2625ex 411 . . . . . 6 ({𝐵} = ∅ → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2714, 26sylbi 216 . . . . 5 𝐵 ∈ V → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2813, 27syl5 34 . . . 4 𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2928adantrd 490 . . 3 𝐵 ∈ V → ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
3012, 29pm2.61i 182 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
315, 30impbid 211 1 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  Vcvv 3462  c0 4325  {csn 4633   × cxp 5682  ran crn 5685  Rel wrel 5689   Fn wfn 6551  wf 6552  ontowfo 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fo 6562  df-fv 6564
This theorem is referenced by:  imadrhmcl  20778  nvo00  30697  zar0ring  33695  esumnul  33883  esum0  33884  volsupnfl  37368
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