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Theorem fconstfvm 5647
 Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5646. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconstfvm (∃𝑦 𝑦𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑦,𝐴
Allowed substitution hints:   𝐵(𝑦)   𝐹(𝑦)

Proof of Theorem fconstfvm
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5281 . . 3 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2 fvconst 5617 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
32ralrimiva 2509 . . 3 (𝐹:𝐴⟶{𝐵} → ∀𝑥𝐴 (𝐹𝑥) = 𝐵)
41, 3jca 304 . 2 (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
5 fvelrnb 5478 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝐴 (𝐹𝑧) = 𝑤))
6 fveq2 5430 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
76eqeq1d 2149 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝑧) = 𝐵))
87rspccva 2793 . . . . . . . . . . . 12 ((∀𝑥𝐴 (𝐹𝑥) = 𝐵𝑧𝐴) → (𝐹𝑧) = 𝐵)
98eqeq1d 2149 . . . . . . . . . . 11 ((∀𝑥𝐴 (𝐹𝑥) = 𝐵𝑧𝐴) → ((𝐹𝑧) = 𝑤𝐵 = 𝑤))
109rexbidva 2436 . . . . . . . . . 10 (∀𝑥𝐴 (𝐹𝑥) = 𝐵 → (∃𝑧𝐴 (𝐹𝑧) = 𝑤 ↔ ∃𝑧𝐴 𝐵 = 𝑤))
11 r19.9rmv 3460 . . . . . . . . . . 11 (∃𝑦 𝑦𝐴 → (𝐵 = 𝑤 ↔ ∃𝑧𝐴 𝐵 = 𝑤))
1211bicomd 140 . . . . . . . . . 10 (∃𝑦 𝑦𝐴 → (∃𝑧𝐴 𝐵 = 𝑤𝐵 = 𝑤))
1310, 12sylan9bbr 459 . . . . . . . . 9 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → (∃𝑧𝐴 (𝐹𝑧) = 𝑤𝐵 = 𝑤))
145, 13sylan9bbr 459 . . . . . . . 8 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹𝐵 = 𝑤))
15 velsn 3550 . . . . . . . . 9 (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16 eqcom 2142 . . . . . . . . 9 (𝑤 = 𝐵𝐵 = 𝑤)
1715, 16bitr2i 184 . . . . . . . 8 (𝐵 = 𝑤𝑤 ∈ {𝐵})
1814, 17syl6bb 195 . . . . . . 7 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹𝑤 ∈ {𝐵}))
1918eqrdv 2138 . . . . . 6 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
2019an32s 558 . . . . 5 (((∃𝑦 𝑦𝐴𝐹 Fn 𝐴) ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → ran 𝐹 = {𝐵})
2120exp31 362 . . . 4 (∃𝑦 𝑦𝐴 → (𝐹 Fn 𝐴 → (∀𝑥𝐴 (𝐹𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
2221imdistand 444 . . 3 (∃𝑦 𝑦𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23 df-fo 5138 . . . 4 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24 fof 5354 . . . 4 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
2523, 24sylbir 134 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
2622, 25syl6 33 . 2 (∃𝑦 𝑦𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → 𝐹:𝐴⟶{𝐵}))
274, 26impbid2 142 1 (∃𝑦 𝑦𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {csn 3533  ran crn 4549   Fn wfn 5127  ⟶wf 5128  –onto→wfo 5130  ‘cfv 5132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fo 5138  df-fv 5140 This theorem is referenced by:  fconst3m  5648
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