ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1veqaeq GIF version

Theorem f1veqaeq 5748
Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1veqaeq ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1veqaeq
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 5747 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)))
2 fveq2 5496 . . . . . . . 8 (𝑐 = 𝐶 → (𝐹𝑐) = (𝐹𝐶))
32eqeq1d 2179 . . . . . . 7 (𝑐 = 𝐶 → ((𝐹𝑐) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝑑)))
4 eqeq1 2177 . . . . . . 7 (𝑐 = 𝐶 → (𝑐 = 𝑑𝐶 = 𝑑))
53, 4imbi12d 233 . . . . . 6 (𝑐 = 𝐶 → (((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑)))
6 fveq2 5496 . . . . . . . 8 (𝑑 = 𝐷 → (𝐹𝑑) = (𝐹𝐷))
76eqeq2d 2182 . . . . . . 7 (𝑑 = 𝐷 → ((𝐹𝐶) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝐷)))
8 eqeq2 2180 . . . . . . 7 (𝑑 = 𝐷 → (𝐶 = 𝑑𝐶 = 𝐷))
97, 8imbi12d 233 . . . . . 6 (𝑑 = 𝐷 → (((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
105, 9rspc2v 2847 . . . . 5 ((𝐶𝐴𝐷𝐴) → (∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1110com12 30 . . . 4 (∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1211adantl 275 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)) → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
131, 12sylbi 120 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1413imp 123 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  wral 2448  wf 5194  1-1wf1 5195  cfv 5198
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-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206
This theorem is referenced by:  f1fveq  5751  f1ocnvfvrneq  5761  f1o2ndf1  6207  fidceq  6847  difinfsnlem  7076  difinfsn  7077  iseqf1olemab  10445  iseqf1olemnanb  10446  pwle2  14031
  Copyright terms: Public domain W3C validator