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Theorem fconst5 6842
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 5695 . . . 4 (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = ran (𝐴 × {𝐵}))
2 rnxp 5910 . . . . 5 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
32eqeq2d 2807 . . . 4 (𝐴 ≠ ∅ → (ran 𝐹 = ran (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
41, 3syl5ib 245 . . 3 (𝐴 ≠ ∅ → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
54adantl 482 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
6 df-fo 6238 . . . . . . 7 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
7 fof 6465 . . . . . . 7 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
86, 7sylbir 236 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
9 fconst2g 6839 . . . . . 6 (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
108, 9syl5ib 245 . . . . 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 6331 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
14 snprc 4566 . . . . . 6 𝐵 ∈ V ↔ {𝐵} = ∅)
15 relrn0 5728 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
1615biimprd 249 . . . . . . . . 9 (Rel 𝐹 → (ran 𝐹 = ∅ → 𝐹 = ∅))
1716adantl 482 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = ∅ → 𝐹 = ∅))
18 eqeq2 2808 . . . . . . . . 9 ({𝐵} = ∅ → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
1918adantr 481 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
20 xpeq2 5471 . . . . . . . . . . 11 ({𝐵} = ∅ → (𝐴 × {𝐵}) = (𝐴 × ∅))
21 xp0 5898 . . . . . . . . . . 11 (𝐴 × ∅) = ∅
2220, 21syl6eq 2849 . . . . . . . . . 10 ({𝐵} = ∅ → (𝐴 × {𝐵}) = ∅)
2322eqeq2d 2807 . . . . . . . . 9 ({𝐵} = ∅ → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2423adantr 481 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2517, 19, 243imtr4d 295 . . . . . . 7 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
2625ex 413 . . . . . 6 ({𝐵} = ∅ → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2714, 26sylbi 218 . . . . 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 183 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
315, 30impbid 213 1 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wne 2986  Vcvv 3440  c0 4217  {csn 4478   × cxp 5448  ran crn 5451  Rel wrel 5455   Fn wfn 6227  wf 6228  ontowfo 6230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fo 6238  df-fv 6240
This theorem is referenced by:  rnmptc  6843  nvo00  28225  esumnul  30920  esum0  30921  volsupnfl  34489
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