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Theorem fconst5 7206
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 5935 . . . 4 (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = ran (𝐴 × {𝐵}))
2 rnxp 6169 . . . . 5 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
32eqeq2d 2743 . . . 4 (𝐴 ≠ ∅ → (ran 𝐹 = ran (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
41, 3imbitrid 243 . . 3 (𝐴 ≠ ∅ → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
54adantl 482 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
6 df-fo 6549 . . . . . . 7 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
7 fof 6805 . . . . . . 7 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
86, 7sylbir 234 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
9 fconst2g 7203 . . . . . 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 416 . . . 4 (𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
1211adantrd 492 . . 3 (𝐵 ∈ V → ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
13 fnrel 6651 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
14 snprc 4721 . . . . . 6 𝐵 ∈ V ↔ {𝐵} = ∅)
15 relrn0 5968 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
1615biimprd 247 . . . . . . . . 9 (Rel 𝐹 → (ran 𝐹 = ∅ → 𝐹 = ∅))
1716adantl 482 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = ∅ → 𝐹 = ∅))
18 eqeq2 2744 . . . . . . . . 9 ({𝐵} = ∅ → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
1918adantr 481 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
20 xpeq2 5697 . . . . . . . . . . 11 ({𝐵} = ∅ → (𝐴 × {𝐵}) = (𝐴 × ∅))
21 xp0 6157 . . . . . . . . . . 11 (𝐴 × ∅) = ∅
2220, 21eqtrdi 2788 . . . . . . . . . 10 ({𝐵} = ∅ → (𝐴 × {𝐵}) = ∅)
2322eqeq2d 2743 . . . . . . . . 9 ({𝐵} = ∅ → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2423adantr 481 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2517, 19, 243imtr4d 293 . . . . . . 7 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
2625ex 413 . . . . . 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 492 . . 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 396   = wceq 1541  wcel 2106  wne 2940  Vcvv 3474  c0 4322  {csn 4628   × cxp 5674  ran crn 5677  Rel wrel 5681   Fn wfn 6538  wf 6539  ontowfo 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551
This theorem is referenced by:  rnmptcOLD  7208  imadrhmcl  20412  nvo00  30009  zar0ring  32853  esumnul  33041  esum0  33042  volsupnfl  36528
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