Theorem fcoresf1b 47664

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 484	. . . 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 484	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 488	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresf1 47663	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
11	10	ex 416	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
12		f1co 6773	. . . 4 ⊢ ((𝑌:𝐸–1-1→𝐷 ∧ 𝑋:𝑃–1-1→𝐸) → (𝑌 ∘ 𝑋):𝑃–1-1→𝐷)
13	12	ancoms 462	. . 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 47661	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
15		f1eq1 6755	. . . 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 248	. 2 ⊢ (𝜑 → ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷))
18	11, 17	impbid 214	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∩ cin 3903  ^◡ccnv 5646  ran crn 5648   ↾ cres 5649   “ cima 5650   ∘ ccom 5651  ⟶wf 6517  –1-1→wf1 6518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-fv 6529

This theorem is referenced by:  fcoresf1ob  47667