Theorem fcoreslem2 47060

Description: Lemma 2 for fcores 47063. (Contributed by AV, 17-Sep-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)

Assertion

Ref	Expression
fcoreslem2	⊢ (𝜑 → ran 𝑋 = 𝐸)

Proof of Theorem fcoreslem2

Step	Hyp	Ref	Expression
1		df-ima 5672	. . 3 ⊢ (𝐹 “ 𝑃) = ran (𝐹 ↾ 𝑃)
2		fcores.x	. . . . . 6 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
3	2	rneqi 5922	. . . . 5 ⊢ ran 𝑋 = ran (𝐹 ↾ 𝑃)
4	3	eqcomi 2745	. . . 4 ⊢ ran (𝐹 ↾ 𝑃) = ran 𝑋
5	4	a1i 11	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝑃) = ran 𝑋)
6	1, 5	eqtr2id 2784	. 2 ⊢ (𝜑 → ran 𝑋 = (𝐹 “ 𝑃))
7		fcores.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		fcores.e	. . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
9		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
10	7, 8, 9	fcoreslem1 47059	. . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
11	10	imaeq2d 6052	. 2 ⊢ (𝜑 → (𝐹 “ 𝑃) = (𝐹 “ (◡𝐹 “ 𝐸)))
12	7	ffund 6715	. . . 4 ⊢ (𝜑 → Fun 𝐹)
13		funimacnv 6622	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
15		inss1 4217	. . . . . 6 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ ran 𝐹
16	8, 15	eqsstri 4010	. . . . 5 ⊢ 𝐸 ⊆ ran 𝐹
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ran 𝐹)
18		dfss2 3949	. . . 4 ⊢ (𝐸 ⊆ ran 𝐹 ↔ (𝐸 ∩ ran 𝐹) = 𝐸)
19	17, 18	sylib 218	. . 3 ⊢ (𝜑 → (𝐸 ∩ ran 𝐹) = 𝐸)
20	14, 19	eqtrd 2771	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = 𝐸)
21	6, 11, 20	3eqtrd 2775	1 ⊢ (𝜑 → ran 𝑋 = 𝐸)

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3930   ⊆ wss 3931  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6530  ⟶wf 6532

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-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  fcoreslem4  47062  fcoresf1  47065