Theorem fcoresfob 47535

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 481	. . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝐺:𝐶⟶𝐷)
8		fcores.y	. . 3 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
9		simpr 485	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
10	2, 3, 4, 5, 7, 8, 9	fcoresfo 47534	. 2 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
11	1, 3, 4, 5	fcoreslem3 47528	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
12	11	anim1ci 622	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸))
13		foco 6753	. . . 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 47530	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
17		foeq1 6735	. . . 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 258	. 2 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)
20	10, 19	impbida 806	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∩ cin 3882  ^◡ccnv 5617  ran crn 5619   ↾ cres 5620   “ cima 5621   ∘ ccom 5622  ⟶wf 6481  –onto→wfo 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493

This theorem is referenced by:  fcoresf1ob  47536