Theorem f1eq123d 6802

Description: Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
f1eq123d.1	⊢ (𝜑 → 𝐹 = 𝐺)
f1eq123d.2	⊢ (𝜑 → 𝐴 = 𝐵)
f1eq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
f1eq123d	⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))

Proof of Theorem f1eq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		f1eq1 6759	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		f1eq2 6760	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		f1eq3 6761	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
10	3, 6, 9	3bitrd 308	1 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563  –1-1→wf1 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530

This theorem is referenced by:  f10d  6845  fthf1  17964  cofth  17982  rngqiprngimf1  21399  istrkgld  28682  istrkg2ld  28683  isushgr  29316  isuspgr  29407  isusgr  29408  isuspgrop  29416  isusgrop  29417  ausgrusgrb  29420  ausgrusgri  29423  usgrstrrepe  29490  uspgr1e  29499  usgrres1  29570  usgrexi  29696  uspgr2wlkeq  29900  usgr2trlncl  30014  aciunf1  32916  pfxf1  33170  s1f1  33171  tocycfv  33337  tocycf  33345  tocyc01  33346  cycpmco2f1  33352  cycpmco2rn  33353  cycpmco2lem1  33354  cycpmco2lem2  33355  cycpmco2lem3  33356  cycpmco2lem4  33357  cycpmco2lem5  33358  cycpmco2lem6  33359  cycpmco2lem7  33360  cycpmco2  33361  cycpm3cl2  33364  cycpmconjv  33370  tocyccntz  33372  cyc3evpm  33378  cycpmgcl  33381  cycpmconjslem2  33383  cyc3conja  33385  dimkerim  33929  f1resfz0f1d  35471  aks6d1c2  42754  f1cof1b  47670  fundcmpsurinjALT  48017  upgrimtrls  48527  stgrusgra  48580  gpgusgra  48678  cofidf2  49750