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Theorem f1veqaeq 6992
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 6990 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)))
2 fveqeq2 6655 . . . . . 6 (𝑐 = 𝐶 → ((𝐹𝑐) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝑑)))
3 eqeq1 2824 . . . . . 6 (𝑐 = 𝐶 → (𝑐 = 𝑑𝐶 = 𝑑))
42, 3imbi12d 347 . . . . 5 (𝑐 = 𝐶 → (((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑)))
5 fveq2 6646 . . . . . . 7 (𝑑 = 𝐷 → (𝐹𝑑) = (𝐹𝐷))
65eqeq2d 2831 . . . . . 6 (𝑑 = 𝐷 → ((𝐹𝐶) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝐷)))
7 eqeq2 2832 . . . . . 6 (𝑑 = 𝐷 → (𝐶 = 𝑑𝐶 = 𝐷))
86, 7imbi12d 347 . . . . 5 (𝑑 = 𝐷 → (((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
94, 8rspc2v 3612 . . . 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 507 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1211imp 409 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3125  wf 6327  1-1wf1 6328  cfv 6331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fv 6339
This theorem is referenced by:  f1cofveqaeq  6993  f1cofveqaeqALT  6994  2f1fvneq  6995  f1fveq  6997  f1prex  7017  f1ocnvfvrneq  7019  f1o2ndf1  7796  symgfvne  18488  f1ghm0to0  19471  mat2pmatf1  21313  f1otrg  26644  uspgr2wlkeq  27414  pthdivtx  27497  spthdep  27502  spthonepeq  27520  usgr2trlncl  27528  ccatf1  30612  swrdf1  30617  cycpmrn  30793  f1resveqaeq  32366  poimirlem1  34934  poimirlem9  34942  poimirlem22  34955  mblfinlem2  34971  isomuspgrlem1  44137
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