Theorem f1oeq123d 5066

Description: Equality deduction for one-to-one onto 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
f1oeq123d	⊢ (φ → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷))

Proof of Theorem f1oeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 ⊢ (φ → 𝐹 = 𝐺)
2		f1oeq1 5060	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:A–1-1-onto→𝐶))
3	1, 2	syl 14	. 2 ⊢ (φ → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:A–1-1-onto→𝐶))
4		f1eq123d.2	. . 3 ⊢ (φ → A = B)
5		f1oeq2 5061	. . 3 ⊢ (A = B → (𝐺:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐶))
6	4, 5	syl 14	. 2 ⊢ (φ → (𝐺:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐶))
7		f1eq123d.3	. . 3 ⊢ (φ → 𝐶 = 𝐷)
8		f1oeq3 5062	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:B–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷))
9	7, 8	syl 14	. 2 ⊢ (φ → (𝐺:B–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷))
10	3, 6, 9	3bitrd 203	1 ⊢ (φ → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  –1-1-onto→wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  f1oprg  5111