Theorem f1eq123d 6795

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 6754	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		f1eq2 6755	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		f1eq3 6756	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
10	3, 6, 9	3bitrd 305	1 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  –1-1→wf1 6511

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-ext 2702

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-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519

This theorem is referenced by:  f10d  6837  fthf1  17888  cofth  17906  rngqiprngimf1  21217  istrkgld  28393  istrkg2ld  28394  isushgr  28995  isuspgr  29086  isusgr  29087  isuspgrop  29095  isusgrop  29096  ausgrusgrb  29099  ausgrusgri  29102  usgrstrrepe  29169  uspgr1e  29178  usgrres1  29249  usgrexi  29375  uspgr2wlkeq  29581  usgr2trlncl  29697  aciunf1  32594  pfxf1  32870  s1f1  32871  tocycfv  33073  tocycf  33081  tocyc01  33082  cycpmco2f1  33088  cycpmco2rn  33089  cycpmco2lem1  33090  cycpmco2lem2  33091  cycpmco2lem3  33092  cycpmco2lem4  33093  cycpmco2lem5  33094  cycpmco2lem6  33095  cycpmco2lem7  33096  cycpmco2  33097  cycpm3cl2  33100  cycpmconjv  33106  tocyccntz  33108  cyc3evpm  33114  cycpmgcl  33117  cycpmconjslem2  33119  cyc3conja  33121  dimkerim  33630  f1resfz0f1d  35108  aks6d1c2  42125  f1cof1b  47082  fundcmpsurinjALT  47417  upgrimtrls  47910  stgrusgra  47962  gpgusgra  48052  cofidf2  49113