Theorem fcoresf1ob 44567

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 44564	. . . 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 44566	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
9	7, 8	anbi12d 631	. . 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 469	. . 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 287	. 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 6440	. 2 ⊢ ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
13		df-f1o 6440	. . 3 ⊢ (𝑌:𝐸–1-1-onto→𝐷 ↔ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷))
14	13	anbi2i 623	. 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 314	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∩ cin 3886  ^◡ccnv 5588  ran crn 5590   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  ⟶wf 6429  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by: (None)