Theorem f1ocof1ob2 47087

Description: If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. Symmetric version of f1ocof1ob 47086 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)

Assertion

Ref	Expression
f1ocof1ob2	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))

Proof of Theorem f1ocof1ob2

Step	Hyp	Ref	Expression
1		f1ocof1ob 47086	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
2		f1f1orn 6814	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐶 → 𝐹:𝐴–1-1-onto→ran 𝐹)
3		f1oeq3 6793	. . . . . 6 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ 𝐹:𝐴–1-1-onto→𝐶))
4	2, 3	imbitrid 244	. . . . 5 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴–1-1→𝐶 → 𝐹:𝐴–1-1-onto→𝐶))
5	4	3ad2ant3 1135	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→𝐶 → 𝐹:𝐴–1-1-onto→𝐶))
6		f1of1 6802	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐶 → 𝐹:𝐴–1-1→𝐶)
7	5, 6	impbid1 225	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐴–1-1-onto→𝐶))
8	7	anbi1d 631	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷) ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
9	1, 8	bitrd 279	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ran crn 5642   ∘ ccom 5645  ⟶wf 6510  –1-1→wf1 6511  –1-1-onto→wf1o 6513

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by: (None)