Theorem fcoresf1b 48345

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⟶1-1𝐷 ↔ (𝑋:𝑃⟶1-1𝐸 ∧ 𝑌:𝐸⟶1-1𝐷)))

Proof of Theorem fcoresf1b

Step	Hyp	Ref	Expression
1		fcores.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃⟶1-1𝐷) → 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃⟶1-1𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 490	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃⟶1-1𝐷) → (𝐺 ∘ 𝐹):𝑃⟶1-1𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresf1 48344	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃⟶1-1𝐷) → (𝑋:𝑃⟶1-1𝐸 ∧ 𝑌:𝐸⟶1-1𝐷))
11	10	ex 418	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃⟶1-1𝐷 → (𝑋:𝑃⟶1-1𝐸 ∧ 𝑌:𝐸⟶1-1𝐷)))
12		f1co 6813	. . . 4 ⊢ ((𝑌:𝐸⟶1-1𝐷 ∧ 𝑋:𝑃⟶1-1𝐸) → (𝑌 ∘ 𝑋):𝑃⟶1-1𝐷)
13	12	ancoms 464	. . 3 ⊢ ((𝑋:𝑃⟶1-1𝐸 ∧ 𝑌:𝐸⟶1-1𝐷) → (𝑌 ∘ 𝑋):𝑃⟶1-1𝐷)
14	1, 3, 4, 5, 6, 8	fcores 48342	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
15		f1eq1 6795	. . . 4 ⊢ ((𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋) → ((𝐺 ∘ 𝐹):𝑃⟶1-1𝐷 ↔ (𝑌 ∘ 𝑋):𝑃⟶1-1𝐷))
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃⟶1-1𝐷 ↔ (𝑌 ∘ 𝑋):𝑃⟶1-1𝐷))
17	13, 16	imbitrrid 249	. 2 ⊢ (𝜑 → ((𝑋:𝑃⟶1-1𝐸 ∧ 𝑌:𝐸⟶1-1𝐷) → (𝐺 ∘ 𝐹):𝑃⟶1-1𝐷))
18	11, 17	impbid 215	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃⟶1-1𝐷 ↔ (𝑋:𝑃⟶1-1𝐸 ∧ 𝑌:𝐸⟶1-1𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∩ cin 4092  ^◡ccnv 5654  ran crn 5656   ↾ cres 5657   “ cima 5658   ∘ ccom 5659  ⟶wf 6554  ⟶1-1wf1 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-fv 6566

This theorem is used by:  fcoresf1ob  48348