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Theorem f1veqaeq 6168
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

Proof of Theorem f1veqaeq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 6166 . . 3
2 fveq2 5871 . . . . . . . 8
32eqeq1d 2459 . . . . . . 7
4 eqeq1 2461 . . . . . . 7
53, 4imbi12d 320 . . . . . 6
6 fveq2 5871 . . . . . . . 8
76eqeq2d 2471 . . . . . . 7
8 eqeq2 2472 . . . . . . 7
97, 8imbi12d 320 . . . . . 6
105, 9rspc2v 3219 . . . . 5
1110com12 31 . . . 4
1211adantl 466 . . 3
131, 12sylbi 195 . 2
1413imp 429 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  -->wf 5589  -1-1->wf1 5590  `cfv 5593
This theorem is referenced by:  f1fveq  6170  f1ocnvfvrneq  6189  f1o2ndf1  6908  symgfvne  16413  f1rhm0to0  17389  mat2pmatf1  19230  f1otrg  24174  spthonepeq  24589  4cycl4dv  24667  usg2wlkeq  24708  mblfinlem2  30052  2f1fvneq  32307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fv 5601
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