Theorem fcoresfob 48281

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 486	. . 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 486	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . 3 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 490	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresfo 48280	. 2 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
11	1, 3, 4, 5	fcoreslem3 48274	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
12	11	anim1ci 628	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸))
13		foco 6818	. . . 4 ⊢ ((𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸) → (𝑌 ∘ 𝑋):𝑃–onto→𝐷)
14	12, 13	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌 ∘ 𝑋):𝑃–onto→𝐷)
15	1, 3, 4, 5, 6, 8	fcores 48276	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
16	15	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
17		foeq1 6800	. . . 4 ⊢ ((𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋) → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–onto→𝐷))
18	16, 17	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–onto→𝐷))
19	14, 18	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
20	10, 19	impbida 813	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∩ cin 4091  ^◡ccnv 5653  ran crn 5655   ↾ cres 5656   “ cima 5657   ∘ ccom 5658  ⟶wf 6542  –onto→wfo 6544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fo 6552  df-fv 6554

This theorem is used by:  fcoresf1ob  48282