Theorem fcoresf1b 44564

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 481	. . . 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 481	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 485	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresf1 44563	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
11	10	ex 413	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
12		f1co 6682	. . . 4 ⊢ ((𝑌:𝐸–1-1→𝐷 ∧ 𝑋:𝑃–1-1→𝐸) → (𝑌 ∘ 𝑋):𝑃–1-1→𝐷)
13	12	ancoms 459	. . 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 44561	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
15		f1eq1 6665	. . . 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 396   = wceq 1539   ∩ cin 3886  ^◡ccnv 5588  ran crn 5590   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  ⟶wf 6429  –1-1→wf1 6430

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-fv 6441

This theorem is referenced by:  fcoresf1ob  44567