Theorem fcoresf1b 47082

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 47081	. . 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 6815	. . . 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 47079	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
15		f1eq1 6799	. . . 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	imbitrrid 246	. 2 ⊢ (𝜑 → ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷))
18	11, 17	impbid 212	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∩ cin 3950  ^◡ccnv 5684  ran crn 5686   ↾ cres 5687   “ cima 5688   ∘ ccom 5689  ⟶wf 6557  –1-1→wf1 6558

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-fv 6569

This theorem is referenced by:  fcoresf1ob  47085