Theorem f1ocof1ob 47838

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 6719	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2	1	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶ran 𝐹)
3		feq3 6685	. . . . . . 7 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐶))
4	3	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐶))
5	2, 4	mpbid 235	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶𝐶)
6		f1cof1b 47834	. . . . 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 1437	. . . 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 6705	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
9		fnfocofob 47836	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
10	8, 9	syl3an1 1181	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
11	7, 10	anbi12d 643	. . 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 473	. . 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 290	. 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 6543	. 2 ⊢ ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝐴–onto→𝐷))
15		df-f1o 6543	. . 3 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐺:𝐶–onto→𝐷))
16	15	anbi2i 634	. 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 317	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ran crn 5662   ∘ ccom 5665   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  f1ocof1ob2  47839