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Theorem f1veqaeq 7277
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 7275 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)))
2 fveqeq2 6915 . . . . . 6 (𝑐 = 𝐶 → ((𝐹𝑐) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝑑)))
3 eqeq1 2741 . . . . . 6 (𝑐 = 𝐶 → (𝑐 = 𝑑𝐶 = 𝑑))
42, 3imbi12d 344 . . . . 5 (𝑐 = 𝐶 → (((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑)))
5 fveq2 6906 . . . . . . 7 (𝑑 = 𝐷 → (𝐹𝑑) = (𝐹𝐷))
65eqeq2d 2748 . . . . . 6 (𝑑 = 𝐷 → ((𝐹𝐶) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝐷)))
7 eqeq2 2749 . . . . . 6 (𝑑 = 𝐷 → (𝐶 = 𝑑𝐶 = 𝐷))
86, 7imbi12d 344 . . . . 5 (𝑑 = 𝐷 → (((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
94, 8rspc2v 3633 . . . 4 ((𝐶𝐴𝐷𝐴) → (∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
109com12 32 . . 3 (∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
111, 10simplbiim 504 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1211imp 406 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wf 6557  1-1wf1 6558  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  f1cofveqaeq  7278  f1cofveqaeqALT  7279  2f1fvneq  7280  f1fveq  7282  f1cdmsn  7302  f1prex  7304  f1ocnvfvrneq  7306  fvf1pr  7327  f1o2ndf1  8147  fvf1tp  13829  f1ghm0to0  19263  symgfvne  19398  mat2pmatf1  22735  f1otrg  28879  uspgr2wlkeq  29664  pthdivtx  29747  spthdep  29754  spthonepeq  29772  usgr2trlncl  29780  ccatf1  32933  swrdf1  32941  cycpmrn  33163  f1resveqaeq  35099  poimirlem1  37628  poimirlem9  37636  poimirlem22  37649  mblfinlem2  37665  ricdrng1  42538  isuspgrim0lem  47871  isubgr3stgrlem7  47939
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