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Theorem f1veqaeq 6395
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 6393 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)))
2 fveq2 6087 . . . . . . . 8 (𝑐 = 𝐶 → (𝐹𝑐) = (𝐹𝐶))
32eqeq1d 2611 . . . . . . 7 (𝑐 = 𝐶 → ((𝐹𝑐) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝑑)))
4 eqeq1 2613 . . . . . . 7 (𝑐 = 𝐶 → (𝑐 = 𝑑𝐶 = 𝑑))
53, 4imbi12d 332 . . . . . 6 (𝑐 = 𝐶 → (((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑)))
6 fveq2 6087 . . . . . . . 8 (𝑑 = 𝐷 → (𝐹𝑑) = (𝐹𝐷))
76eqeq2d 2619 . . . . . . 7 (𝑑 = 𝐷 → ((𝐹𝐶) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝐷)))
8 eqeq2 2620 . . . . . . 7 (𝑑 = 𝐷 → (𝐶 = 𝑑𝐶 = 𝐷))
97, 8imbi12d 332 . . . . . 6 (𝑑 = 𝐷 → (((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
105, 9rspc2v 3292 . . . . 5 ((𝐶𝐴𝐷𝐴) → (∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1110com12 32 . . . 4 (∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1211adantl 480 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑐𝐴𝑑𝐴 ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)) → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
131, 12sylbi 205 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶𝐴𝐷𝐴) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1413imp 443 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  wf 5785  1-1wf1 5786  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fv 5797
This theorem is referenced by:  f1fveq  6397  f1prex  6416  f1ocnvfvrneq  6418  f1o2ndf1  7149  symgfvne  17579  f1rhm0to0  18511  mat2pmatf1  20300  f1otrg  25496  spthonepeq  25910  4cycl4dv  25988  usg2wlkeq  26029  poimirlem1  32363  poimirlem9  32371  poimirlem22  32384  mblfinlem2  32400  2f1fvneq  40117  f1cofveqaeq  40118  f1cofveqaeqALT  40119  uspgr2wlkeq  40835  pthdivtx  40916  spthdep  40921  spthonepeq-av  40939  usgr2trlncl  40947
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