Theorem f1veqaeq 7213

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 7211	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 ((𝐹‘𝑐) = (𝐹‘𝑑) → 𝑐 = 𝑑)))
2		fveqeq2 6849	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝑑)))
3		eqeq1 2733	. . . . . 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 6840	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝐹‘𝑑) = (𝐹‘𝐷))
6	5	eqeq2d 2740	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝐹‘𝐶) = (𝐹‘𝑑) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
7		eqeq2 2741	. . . . . 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 3596	. . . 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 1540   ∈ wcel 2109  ∀wral 3044  ⟶wf 6495  –1-1→wf1 6496  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fv 6507

This theorem is referenced by:  f1cofveqaeq  7214  f1cofveqaeqALT  7215  dff14i  7216  f1fveq  7219  f1cdmsn  7239  f1prex  7241  f1ocnvfvrneq  7243  fvf1pr  7264  f1o2ndf1  8078  fvf1tp  13727  f1ghm0to0  19159  symgfvne  19295  mat2pmatf1  22649  f1otrg  28851  uspgr2wlkeq  29626  pthdivtx  29707  spthdep  29714  spthonepeq  29732  usgr2trlncl  29740  ccatf1  32920  swrdf1  32928  cycpmrn  33115  f1resveqaeq  35068  vonf1owev  35088  poimirlem1  37608  poimirlem9  37616  poimirlem22  37629  mblfinlem2  37645  ricdrng1  42509  permaxext  44988  isuspgrim0lem  47886  isubgr3stgrlem7  47964