Theorem f1eq123d 5629

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 5591	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		f1eq2 5592	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶))
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		f1eq3 5593	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
10	3, 6, 9	3bitrd 214	1 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  –1-1→wf1 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380

This theorem is referenced by:  f10d  5673  isushgrm  16296  isuspgren  16381  isusgren  16382  isuspgropen  16388  isusgropen  16389  ausgrusgrben  16392  ausgrusgrien  16395  usgrstrrepeen  16455  uspgr1edc  16464  uspgr2wlkeq  16589