Theorem fcoresf1ob 47870

Description: A composition is bijective iff the restriction of its first component to the minimum domain is bijective and the restriction of its second component to the minimum domain is injective. (Contributed by GL and AV, 7-Oct-2024.)

Hypotheses

Ref	Expression
fcores.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcores.e	⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
fcores.p	⊢ 𝑃 = (◡𝐹 “ 𝐶)
fcores.x	⊢ 𝑋 = (𝐹 ↾ 𝑃)
fcores.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
fcores.y	⊢ 𝑌 = (𝐺 ↾ 𝐸)

Assertion

Ref	Expression
fcoresf1ob	⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))

Proof of Theorem fcoresf1ob

Step	Hyp	Ref	Expression
1		fcores.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		fcores.e	. . . . 5 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
3		fcores.p	. . . . 5 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
4		fcores.x	. . . . 5 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
5		fcores.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
6		fcores.y	. . . . 5 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
7	1, 2, 3, 4, 5, 6	fcoresf1b 47867	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
8	1, 2, 3, 4, 5, 6	fcoresfob 47869	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
9	7, 8	anbi12d 644	. . 3 ⊢ (𝜑 → (((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) ↔ ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) ∧ 𝑌:𝐸–onto→𝐷)))
10		anass 474	. . 3 ⊢ (((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) ∧ 𝑌:𝐸–onto→𝐷) ↔ (𝑋:𝑃–1-1→𝐸 ∧ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷)))
11	9, 10	bitrdi 290	. 2 ⊢ (𝜑 → (((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) ↔ (𝑋:𝑃–1-1→𝐸 ∧ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷))))
12		df-f1o 6547	. 2 ⊢ ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
13		df-f1o 6547	. . 3 ⊢ (𝑌:𝐸–1-1-onto→𝐷 ↔ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷))
14	13	anbi2i 635	. 2 ⊢ ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷) ↔ (𝑋:𝑃–1-1→𝐸 ∧ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷)))
15	11, 12, 14	3bitr4g 317	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∩ cin 3905  ^◡ccnv 5662  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667  ⟶wf 6536  –1-1→wf1 6537  –onto→wfo 6538  –1-1-onto→wf1o 6539

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is used by:  3f1oss1  47872