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:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1veqaeq

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff13 7275	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑)))
2		fveqeq2 6916	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝑑)))
3		eqeq1 2739	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 = 𝑑 ↔ 𝐶 = 𝑑))
4	2, 3	imbi12d 344	. . . . 5 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑)))
5		fveq2 6907	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝐹‘𝑑) = (𝐹‘𝐷))
6	5	eqeq2d 2746	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝐹‘𝐶) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
7		eqeq2 2747	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 = 𝑑 ↔ 𝐶 = 𝐷))
8	6, 7	imbi12d 344	. . . . 5 ⊢ (𝑑 = 𝐷 → (((𝐹‘𝐶) = (𝐹‘𝑑) → 𝐶 = 𝑑) ↔ ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)))
9	4, 8	rspc2v 3633	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)))
10	9	com12 32	. . 3 ⊢ (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)))
11	1, 10	simplbiim 504	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)))
12	11	imp 406	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571

This theorem is referenced by:  f1cofveqaeq  7278  f1cofveqaeqALT  7279  2f1fvneq  7280  f1fveq  7282  f1cdmsn  7302  f1prex  7304  f1ocnvfvrneq  7306  fvf1pr  7327  f1o2ndf1  8146  fvf1tp  13826  f1ghm0to0  19276  symgfvne  19413  mat2pmatf1  22751  f1otrg  28894  uspgr2wlkeq  29679  pthdivtx  29762  spthdep  29767  spthonepeq  29785  usgr2trlncl  29793  ccatf1  32918  swrdf1  32926  cycpmrn  33146  f1resveqaeq  35078  poimirlem1  37608  poimirlem9  37616  poimirlem22  37629  mblfinlem2  37645  ricdrng1  42515  isuspgrim0lem  47809  isubgr3stgrlem7  47875