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Theorem f1veqaeq 7256
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 7254 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)))
2 fveqeq2 6901 . . . . . 6 (𝑐 = 𝐶 → ((𝐹𝑐) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝑑)))
3 eqeq1 2737 . . . . . 6 (𝑐 = 𝐶 → (𝑐 = 𝑑𝐶 = 𝑑))
42, 3imbi12d 345 . . . . 5 (𝑐 = 𝐶 → (((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑)))
5 fveq2 6892 . . . . . . 7 (𝑑 = 𝐷 → (𝐹𝑑) = (𝐹𝐷))
65eqeq2d 2744 . . . . . 6 (𝑑 = 𝐷 → ((𝐹𝐶) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝐷)))
7 eqeq2 2745 . . . . . 6 (𝑑 = 𝐷 → (𝐶 = 𝑑𝐶 = 𝐷))
86, 7imbi12d 345 . . . . 5 (𝑑 = 𝐷 → (((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
94, 8rspc2v 3623 . . . 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 506 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1211imp 408 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  wf 6540  1-1wf1 6541  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552
This theorem is referenced by:  f1cofveqaeq  7257  f1cofveqaeqALT  7258  2f1fvneq  7259  f1fveq  7261  f1cdmsn  7280  f1prex  7282  f1ocnvfvrneq  7284  f1o2ndf1  8108  symgfvne  19248  f1ghm0to0  20279  mat2pmatf1  22231  f1otrg  28122  uspgr2wlkeq  28903  pthdivtx  28986  spthdep  28991  spthonepeq  29009  usgr2trlncl  29017  ccatf1  32115  swrdf1  32120  cycpmrn  32302  f1resveqaeq  34088  poimirlem1  36489  poimirlem9  36497  poimirlem22  36510  mblfinlem2  36526  ricdrng1  41102  isomuspgrlem1  46495
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