Theorem fcoresf1ob 45769

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 45766	. . . 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 45768	. . . 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 286	. 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 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 313	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 1541   ∩ cin 3946  ^◡ccnv 5674  ran crn 5676   ↾ cres 5677   “ cima 5678   ∘ ccom 5679  ⟶wf 6536  –1-1→wf1 6537  –onto→wfo 6538  –1-1-onto→wf1o 6539

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is referenced by: (None)