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Theorem fconstfvm 5406
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5405. (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 5073 . . 3 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2 fvconst 5378 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
32ralrimiva 2409 . . 3 (𝐹:𝐴⟶{𝐵} → ∀𝑥𝐴 (𝐹𝑥) = 𝐵)
41, 3jca 294 . 2 (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
5 fvelrnb 5248 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝐴 (𝐹𝑧) = 𝑤))
6 fveq2 5205 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
76eqeq1d 2064 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝑧) = 𝐵))
87rspccva 2672 . . . . . . . . . . . 12 ((∀𝑥𝐴 (𝐹𝑥) = 𝐵𝑧𝐴) → (𝐹𝑧) = 𝐵)
98eqeq1d 2064 . . . . . . . . . . 11 ((∀𝑥𝐴 (𝐹𝑥) = 𝐵𝑧𝐴) → ((𝐹𝑧) = 𝑤𝐵 = 𝑤))
109rexbidva 2340 . . . . . . . . . 10 (∀𝑥𝐴 (𝐹𝑥) = 𝐵 → (∃𝑧𝐴 (𝐹𝑧) = 𝑤 ↔ ∃𝑧𝐴 𝐵 = 𝑤))
11 r19.9rmv 3340 . . . . . . . . . . 11 (∃𝑦 𝑦𝐴 → (𝐵 = 𝑤 ↔ ∃𝑧𝐴 𝐵 = 𝑤))
1211bicomd 133 . . . . . . . . . 10 (∃𝑦 𝑦𝐴 → (∃𝑧𝐴 𝐵 = 𝑤𝐵 = 𝑤))
1310, 12sylan9bbr 444 . . . . . . . . 9 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → (∃𝑧𝐴 (𝐹𝑧) = 𝑤𝐵 = 𝑤))
145, 13sylan9bbr 444 . . . . . . . 8 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹𝐵 = 𝑤))
15 velsn 3419 . . . . . . . . 9 (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16 eqcom 2058 . . . . . . . . 9 (𝑤 = 𝐵𝐵 = 𝑤)
1715, 16bitr2i 178 . . . . . . . 8 (𝐵 = 𝑤𝑤 ∈ {𝐵})
1814, 17syl6bb 189 . . . . . . 7 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹𝑤 ∈ {𝐵}))
1918eqrdv 2054 . . . . . 6 (((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
2019an32s 510 . . . . 5 (((∃𝑦 𝑦𝐴𝐹 Fn 𝐴) ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → ran 𝐹 = {𝐵})
2120exp31 350 . . . 4 (∃𝑦 𝑦𝐴 → (𝐹 Fn 𝐴 → (∀𝑥𝐴 (𝐹𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
2221imdistand 429 . . 3 (∃𝑦 𝑦𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23 df-fo 4935 . . . 4 (𝐹:𝐴onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24 fof 5133 . . . 4 (𝐹:𝐴onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
2523, 24sylbir 129 . . 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 135 1 (∃𝑦 𝑦𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  {csn 3402  ran crn 4373   Fn wfn 4924  wf 4925  ontowfo 4927  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937
This theorem is referenced by:  fconst3m  5407
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