ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fconst4m GIF version

Theorem fconst4m 5716
Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
fconst4m (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fconst4m
StepHypRef Expression
1 fconst3m 5715 . 2 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
2 cnvimass 4974 . . . . . 6 (𝐹 “ {𝐵}) ⊆ dom 𝐹
3 fndm 5297 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3sseqtrid 3197 . . . . 5 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) ⊆ 𝐴)
54biantrurd 303 . . . 4 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
6 eqss 3162 . . . 4 ((𝐹 “ {𝐵}) = 𝐴 ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
75, 6bitr4di 197 . . 3 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ (𝐹 “ {𝐵}) = 𝐴))
87pm5.32i 451 . 2 ((𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
91, 8bitrdi 195 1 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  wss 3121  {csn 3583  ccnv 4610  dom cdm 4611  cima 4614   Fn wfn 5193  wf 5194
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator