Theorem f1oeq3 5497

Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1oeq3	⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))

Proof of Theorem f1oeq3

Step	Hyp	Ref	Expression
1		f1eq3 5463	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))
2		foeq3 5481	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵))
3	1, 2	anbi12d 473	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴) ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵)))
4		df-f1o 5266	. 2 ⊢ (𝐹:𝐶–1-1-onto→𝐴 ↔ (𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴))
5		df-f1o 5266	. 2 ⊢ (𝐹:𝐶–1-1-onto→𝐵 ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵))
6	3, 4, 5	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  –1-1→wf1 5256  –onto→wfo 5257  –1-1-onto→wf1o 5258

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266

This theorem is referenced by:  f1oeq23  5498  f1oeq123d  5501  f1oeq3d  5504  f1ores  5522  resdif  5529  f1osng  5548  f1oresrab  5730  isoeq5  5855  isoini2  5869  mapsnf1o  6805  bren  6815  xpcomf1o  6893  frechashgf1o  10537  sumeq1  11537  fisumss  11574  fsumcnv  11619  prodeq1f  11734  4sqlem11  12595  ennnfonelemhf1o  12655  ennnfonelemex  12656  ssnnctlemct  12688