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Theorem fconst5 6354
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 5259 . . . 4 (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = ran (𝐴 × {𝐵}))
2 rnxp 5469 . . . . 5 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
32eqeq2d 2619 . . . 4 (𝐴 ≠ ∅ → (ran 𝐹 = ran (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
41, 3syl5ib 232 . . 3 (𝐴 ≠ ∅ → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
54adantl 480 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
6 df-fo 5796 . . . . . . 7 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
7 fof 6013 . . . . . . 7 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
86, 7sylbir 223 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
9 fconst2g 6351 . . . . . 6 (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
108, 9syl5ib 232 . . . . 5 (𝐵 ∈ V → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹 = (𝐴 × {𝐵})))
1110expd 450 . . . 4 (𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
1211adantrd 482 . . 3 (𝐵 ∈ V → ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
13 fnrel 5889 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
14 snprc 4196 . . . . . 6 𝐵 ∈ V ↔ {𝐵} = ∅)
15 relrn0 5291 . . . . . . . . . 10 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
1615biimprd 236 . . . . . . . . 9 (Rel 𝐹 → (ran 𝐹 = ∅ → 𝐹 = ∅))
1716adantl 480 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = ∅ → 𝐹 = ∅))
18 eqeq2 2620 . . . . . . . . 9 ({𝐵} = ∅ → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
1918adantr 479 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
20 xpeq2 5043 . . . . . . . . . . 11 ({𝐵} = ∅ → (𝐴 × {𝐵}) = (𝐴 × ∅))
21 xp0 5457 . . . . . . . . . . 11 (𝐴 × ∅) = ∅
2220, 21syl6eq 2659 . . . . . . . . . 10 ({𝐵} = ∅ → (𝐴 × {𝐵}) = ∅)
2322eqeq2d 2619 . . . . . . . . 9 ({𝐵} = ∅ → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2423adantr 479 . . . . . . . 8 (({𝐵} = ∅ ∧ Rel 𝐹) → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
2517, 19, 243imtr4d 281 . . . . . . 7 (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
2625ex 448 . . . . . 6 ({𝐵} = ∅ → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2714, 26sylbi 205 . . . . 5 𝐵 ∈ V → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2813, 27syl5 33 . . . 4 𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
2928adantrd 482 . . 3 𝐵 ∈ V → ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
3012, 29pm2.61i 174 . 2 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
315, 30impbid 200 1 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  c0 3873  {csn 4124   × cxp 5026  ran crn 5029  Rel wrel 5033   Fn wfn 5785  wf 5786  ontowfo 5788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fo 5796  df-fv 5798
This theorem is referenced by:  nvo00  26806  esumnul  29243  esum0  29244  volsupnfl  32420  rnmptc  38144
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