Theorem fcoresfob 47077

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 47076	. 2 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷)
11	1, 3, 4, 5	fcoreslem3 47070	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
12	11	anim1ci 616	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸))
13		foco 6789	. . . 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 47072	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
17		foeq1 6771	. . . 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 800	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∩ cin 3916  ^◡ccnv 5640  ran crn 5642   ↾ cres 5643   “ cima 5644   ∘ ccom 5645  ⟶wf 6510  –onto→wfo 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522

This theorem is referenced by:  fcoresf1ob  47078