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Theorem fconstfvm 5458
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5457. (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 5117 . . 3 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2 fvconst 5430 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
32ralrimiva 2442 . . 3 (𝐹:𝐴⟶{𝐵} → ∀𝑥𝐴 (𝐹𝑥) = 𝐵)
41, 3jca 300 . 2 (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
5 fvelrnb 5300 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝐴 (𝐹𝑧) = 𝑤))
6 fveq2 5256 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
76eqeq1d 2093 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝑧) = 𝐵))
87rspccva 2713 . . . . . . . . . . . 12 ((∀𝑥𝐴 (𝐹𝑥) = 𝐵𝑧𝐴) → (𝐹𝑧) = 𝐵)
98eqeq1d 2093 . . . . . . . . . . 11 ((∀𝑥𝐴 (𝐹𝑥) = 𝐵𝑧𝐴) → ((𝐹𝑧) = 𝑤𝐵 = 𝑤))
109rexbidva 2373 . . . . . . . . . 10 (∀𝑥𝐴 (𝐹𝑥) = 𝐵 → (∃𝑧𝐴 (𝐹𝑧) = 𝑤 ↔ ∃𝑧𝐴 𝐵 = 𝑤))
11 r19.9rmv 3357 . . . . . . . . . . 11 (∃𝑦 𝑦𝐴 → (𝐵 = 𝑤 ↔ ∃𝑧𝐴 𝐵 = 𝑤))
1211bicomd 139 . . . . . . . . . 10 (∃𝑦 𝑦𝐴 → (∃𝑧𝐴 𝐵 = 𝑤𝐵 = 𝑤))
1310, 12sylan9bbr 451 . . . . . . . . 9 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → (∃𝑧𝐴 (𝐹𝑧) = 𝑤𝐵 = 𝑤))
145, 13sylan9bbr 451 . . . . . . . 8 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹𝐵 = 𝑤))
15 velsn 3442 . . . . . . . . 9 (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16 eqcom 2087 . . . . . . . . 9 (𝑤 = 𝐵𝐵 = 𝑤)
1715, 16bitr2i 183 . . . . . . . 8 (𝐵 = 𝑤𝑤 ∈ {𝐵})
1814, 17syl6bb 194 . . . . . . 7 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹𝑤 ∈ {𝐵}))
1918eqrdv 2083 . . . . . 6 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
2019an32s 533 . . . . 5 (((∃𝑦 𝑦𝐴𝐹 Fn 𝐴) ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → ran 𝐹 = {𝐵})
2120exp31 356 . . . 4 (∃𝑦 𝑦𝐴 → (𝐹 Fn 𝐴 → (∀𝑥𝐴 (𝐹𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
2221imdistand 436 . . 3 (∃𝑦 𝑦𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23 df-fo 4978 . . . 4 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24 fof 5184 . . . 4 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
2523, 24sylbir 133 . . 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 141 1 (∃𝑦 𝑦𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  wral 2355  wrex 2356  {csn 3425  ran crn 4405   Fn wfn 4967  wf 4968  ontowfo 4970  cfv 4972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fo 4978  df-fv 4980
This theorem is referenced by:  fconst3m  5459
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