Theorem f1ocof1ob 48424

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⟶1-1onto𝐷 ↔ (𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷)))

Proof of Theorem f1ocof1ob

Step	Hyp	Ref	Expression
1		ffrn 6753	. . . . . . 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 6719	. . . . . . 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 48420	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1𝐷 ↔ (𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1𝐷)))
7	5, 6	syld3an1 1438	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1𝐷 ↔ (𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1𝐷)))
8		ffn 6739	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
9		fnfocofob 48422	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶onto𝐷 ↔ 𝐺:𝐶⟶onto𝐷))
10	8, 9	syl3an1 1181	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶onto𝐷 ↔ 𝐺:𝐶⟶onto𝐷))
11	7, 10	anbi12d 644	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐺 ∘ 𝐹):𝐴⟶1-1𝐷 ∧ (𝐺 ∘ 𝐹):𝐴⟶onto𝐷) ↔ ((𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1𝐷) ∧ 𝐺:𝐶⟶onto𝐷)))
12		anass 474	. . 3 ⊢ (((𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1𝐷) ∧ 𝐺:𝐶⟶onto𝐷) ↔ (𝐹:𝐴⟶1-1𝐶 ∧ (𝐺:𝐶⟶1-1𝐷 ∧ 𝐺:𝐶⟶onto𝐷)))
13	11, 12	bitrdi 290	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐺 ∘ 𝐹):𝐴⟶1-1𝐷 ∧ (𝐺 ∘ 𝐹):𝐴⟶onto𝐷) ↔ (𝐹:𝐴⟶1-1𝐶 ∧ (𝐺:𝐶⟶1-1𝐷 ∧ 𝐺:𝐶⟶onto𝐷))))
14		df-f1o 6573	. 2 ⊢ ((𝐺 ∘ 𝐹):𝐴⟶1-1onto𝐷 ↔ ((𝐺 ∘ 𝐹):𝐴⟶1-1𝐷 ∧ (𝐺 ∘ 𝐹):𝐴⟶onto𝐷))
15		df-f1o 6573	. . 3 ⊢ (𝐺:𝐶⟶1-1onto𝐷 ↔ (𝐺:𝐶⟶1-1𝐷 ∧ 𝐺:𝐶⟶onto𝐷))
16	15	anbi2i 635	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷) ↔ (𝐹:𝐴⟶1-1𝐶 ∧ (𝐺:𝐶⟶1-1𝐷 ∧ 𝐺:𝐶⟶onto𝐷)))
17	13, 14, 16	3bitr4g 317	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1onto𝐷 ↔ (𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ran crn 5659   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  f1ocof1ob2  48425