Theorem fcoresf1b 44451

Description: A composition is injective iff the restrictions of its components to the minimum domains are 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
fcoresf1b	⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))

Proof of Theorem fcoresf1b

Step	Hyp	Ref	Expression
1		fcores.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → 𝐹:𝐴⟶𝐵)
3		fcores.e	. . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
4		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
5		fcores.x	. . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
6		fcores.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 484	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresf1 44450	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
11	10	ex 412	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
12		f1co 6666	. . . 4 ⊢ ((𝑌:𝐸–1-1→𝐷 ∧ 𝑋:𝑃–1-1→𝐸) → (𝑌 ∘ 𝑋):𝑃–1-1→𝐷)
13	12	ancoms 458	. . 3 ⊢ ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) → (𝑌 ∘ 𝑋):𝑃–1-1→𝐷)
14	1, 3, 4, 5, 6, 8	fcores 44448	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
15		f1eq1 6649	. . . 4 ⊢ ((𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋) → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–1-1→𝐷))
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–1-1→𝐷))
17	13, 16	syl5ibr 245	. 2 ⊢ (𝜑 → ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷))
18	11, 17	impbid 211	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∩ cin 3882  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  ⟶wf 6414  –1-1→wf1 6415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-fv 6426

This theorem is referenced by:  fcoresf1ob  44454