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Theorem fconst5 7207
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 5936 . . . 4 (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = ran (𝐴 × {𝐵}))
2 rnxp 6170 . . . . 5 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
32eqeq2d 2744 . . . 4 (𝐴 ≠ ∅ → (ran 𝐹 = ran (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
41, 3imbitrid 243 . . 3 (𝐴 ≠ ∅ → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
54adantl 483 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
6 df-fo 6550 . . . . . . 7 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
7 fof 6806 . . . . . . 7 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
86, 7sylbir 234 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
9 fconst2g 7204 . . . . . 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 417 . . . 4 (𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
1211adantrd 493 . . 3 (𝐵 ∈ V → ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
13 fnrel 6652 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
14 snprc 4722 . . . . . 6 𝐵 ∈ V ↔ {𝐵} = ∅)
15 relrn0 5969 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
1615biimprd 247 . . . . . . . . 9 (Rel 𝐹 → (ran 𝐹 = ∅ → 𝐹 = ∅))
1716adantl 483 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = ∅ → 𝐹 = ∅))
18 eqeq2 2745 . . . . . . . . 9 ({𝐵} = ∅ → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
1918adantr 482 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
20 xpeq2 5698 . . . . . . . . . . 11 ({𝐵} = ∅ → (𝐴 × {𝐵}) = (𝐴 × ∅))
21 xp0 6158 . . . . . . . . . . 11 (𝐴 × ∅) = ∅
2220, 21eqtrdi 2789 . . . . . . . . . 10 ({𝐵} = ∅ → (𝐴 × {𝐵}) = ∅)
2322eqeq2d 2744 . . . . . . . . 9 ({𝐵} = ∅ → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2423adantr 482 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2517, 19, 243imtr4d 294 . . . . . . 7 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
2625ex 414 . . . . . 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 493 . . 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 397   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  c0 4323  {csn 4629   × cxp 5675  ran crn 5678  Rel wrel 5682   Fn wfn 6539  wf 6540  ontowfo 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552
This theorem is referenced by:  rnmptcOLD  7209  imadrhmcl  20413  nvo00  30014  zar0ring  32858  esumnul  33046  esum0  33047  volsupnfl  36533
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