Theorem f1ocof1ob 47082

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 6701	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2	1	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶ran 𝐹)
3		feq3 6668	. . . . . . 7 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐶))
4	3	3ad2ant3 1135	. . . . . 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 47078	. . . . 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 1412	. . . 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 6688	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
9		fnfocofob 47080	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
10	8, 9	syl3an1 1163	. . . 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 6518	. 2 ⊢ ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝐴–onto→𝐷))
15		df-f1o 6518	. . 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 1540  ran crn 5639   ∘ ccom 5642   Fn wfn 6506  ⟶wf 6507  –1-1→wf1 6508  –onto→wfo 6509  –1-1-onto→wf1o 6510

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  f1ocof1ob2  47083