Theorem fcoresf1ob 46081

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 46078	. . . 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 46080	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
9	7, 8	anbi12d 629	. . 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 467	. . 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 6549	. 2 ⊢ ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
13		df-f1o 6549	. . 3 ⊢ (𝑌:𝐸–1-1-onto→𝐷 ↔ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷))
14	13	anbi2i 621	. 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 394   = wceq 1539   ∩ cin 3946  ^◡ccnv 5674  ran crn 5676   ↾ cres 5677   “ cima 5678   ∘ ccom 5679  ⟶wf 6538  –1-1→wf1 6539  –onto→wfo 6540  –1-1-onto→wf1o 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by: (None)