Theorem fcoresfob 46233

Description: A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (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
fcoresfob	⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))

Proof of Theorem fcoresfob

Step	Hyp	Ref	Expression
1		fcores.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝐹:𝐴⟶𝐵)
3		fcores.e	. . 3 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
4		fcores.p	. . 3 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
5		fcores.x	. . 3 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
6		fcores.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . 3 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresfo 46232	. 2 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
11	1, 3, 4, 5	fcoreslem3 46226	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
12	11	anim1ci 615	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸))
13		foco 6809	. . . 4 ⊢ ((𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸) → (𝑌 ∘ 𝑋):𝑃–onto→𝐷)
14	12, 13	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌 ∘ 𝑋):𝑃–onto→𝐷)
15	1, 3, 4, 5, 6, 8	fcores 46228	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
17		foeq1 6791	. . . 4 ⊢ ((𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋) → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–onto→𝐷))
18	16, 17	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–onto→𝐷))
19	14, 18	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
20	10, 19	impbida 798	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∩ cin 3939  ^◡ccnv 5665  ran crn 5667   ↾ cres 5668   “ cima 5669   ∘ ccom 5670  ⟶wf 6529  –onto→wfo 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541

This theorem is referenced by:  fcoresf1ob  46234