Theorem f1ocof1ob 47031

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. (Contributed by GL and AV, 7-Oct-2024.)

Assertion

Ref	Expression
f1ocof1ob	⊢ ((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)))

Proof of Theorem f1ocof1ob

Step	Hyp	Ref	Expression
1		ffrn 6750	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2	1	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶ran 𝐹)
3		feq3 6719	. . . . . . 7 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐶))
4	3	3ad2ant3 1134	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐶))
5	2, 4	mpbid 232	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶𝐶)
6		f1cof1b 47027	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1→𝐷)))
7	5, 6	syld3an1 1409	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1→𝐷)))
8		ffn 6737	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
9		fnfocofob 47029	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
10	8, 9	syl3an1 1162	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
11	7, 10	anbi12d 632	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝐴–onto→𝐷) ↔ ((𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1→𝐷) ∧ 𝐺:𝐶–onto→𝐷)))
12		anass 468	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1→𝐷) ∧ 𝐺:𝐶–onto→𝐷) ↔ (𝐹:𝐴–1-1→𝐶 ∧ (𝐺:𝐶–1-1→𝐷 ∧ 𝐺:𝐶–onto→𝐷)))
13	11, 12	bitrdi 287	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝐴–onto→𝐷) ↔ (𝐹:𝐴–1-1→𝐶 ∧ (𝐺:𝐶–1-1→𝐷 ∧ 𝐺:𝐶–onto→𝐷))))
14		df-f1o 6570	. 2 ⊢ ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝐴–onto→𝐷))
15		df-f1o 6570	. . 3 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐺:𝐶–onto→𝐷))
16	15	anbi2i 623	. 2 ⊢ ((𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷) ↔ (𝐹:𝐴–1-1→𝐶 ∧ (𝐺:𝐶–1-1→𝐷 ∧ 𝐺:𝐶–onto→𝐷)))
17	13, 14, 16	3bitr4g 314	1 ⊢ ((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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ran crn 5690   ∘ ccom 5693   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  –onto→wfo 6561  –1-1-onto→wf1o 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by:  f1ocof1ob2  47032