Theorem f1ocof1ob 47441

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 6683	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2	1	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶ran 𝐹)
3		feq3 6650	. . . . . . 7 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐶))
4	3	3ad2ant3 1136	. . . . . 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 47437	. . . . 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 1413	. . . 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 6670	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
9		fnfocofob 47439	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
10	8, 9	syl3an1 1164	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷))
11	7, 10	anbi12d 633	. . 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 6507	. 2 ⊢ ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝐴–onto→𝐷))
15		df-f1o 6507	. . 3 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐺:𝐶–onto→𝐷))
16	15	anbi2i 624	. 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 1087   = wceq 1542  ran crn 5633   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496  –1-1→wf1 6497  –onto→wfo 6498  –1-1-onto→wf1o 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508

This theorem is referenced by:  f1ocof1ob2  47442