Theorem f1ocof1ob2 48425

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. Symmetric version of f1ocof1ob 48424 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)

Assertion

Ref	Expression
f1ocof1ob2	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1onto𝐷 ↔ (𝐹:𝐴⟶1-1onto𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷)))

Proof of Theorem f1ocof1ob2

Step	Hyp	Ref	Expression
1		f1ocof1ob 48424	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1onto𝐷 ↔ (𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷)))
2		f1f1orn 6866	. . . . . 6 ⊢ (𝐹:𝐴⟶1-1𝐶 → 𝐹:𝐴⟶1-1ontoran 𝐹)
3		f1oeq3 6844	. . . . . 6 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶1-1ontoran 𝐹 ↔ 𝐹:𝐴⟶1-1onto𝐶))
4	2, 3	imbitrid 247	. . . . 5 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶1-1𝐶 → 𝐹:𝐴⟶1-1onto𝐶))
5	4	3ad2ant3 1153	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴⟶1-1𝐶 → 𝐹:𝐴⟶1-1onto𝐶))
6		f1of1 6853	. . . 4 ⊢ (𝐹:𝐴⟶1-1onto𝐶 → 𝐹:𝐴⟶1-1𝐶)
7	5, 6	impbid1 228	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴⟶1-1𝐶 ↔ 𝐹:𝐴⟶1-1onto𝐶))
8	7	anbi1d 643	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴⟶1-1𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷) ↔ (𝐹:𝐴⟶1-1onto𝐶 ∧ 𝐺:𝐶⟶1-1onto𝐷)))
9	1, 8	bitrd 282	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1onto𝐷 ↔ (𝐹:𝐴⟶1-1onto𝐶 ∧ 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  ⟶wf 6562  ⟶1-1wf1 6563  ⟶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: (None)