Theorem fcoreslem2 47727

Description: Lemma 2 for fcores 47730. (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 5677	. . 3 ⊢ (𝐹 “ 𝑃) = ran (𝐹 ↾ 𝑃)
2		fcores.x	. . . . . 6 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
3	2	rneqi 5930	. . . . 5 ⊢ ran 𝑋 = ran (𝐹 ↾ 𝑃)
4	3	eqcomi 2778	. . . 4 ⊢ ran (𝐹 ↾ 𝑃) = ran 𝑋
5	4	a1i 11	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝑃) = ran 𝑋)
6	1, 5	eqtr2id 2817	. 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 47726	. . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
11	10	imaeq2d 6065	. 2 ⊢ (𝜑 → (𝐹 “ 𝑃) = (𝐹 “ (◡𝐹 “ 𝐸)))
12	7	ffund 6713	. . . 4 ⊢ (𝜑 → Fun 𝐹)
13		funimacnv 6620	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
15		inss1 4197	. . . . . 6 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ ran 𝐹
16	8, 15	eqsstri 3991	. . . . 5 ⊢ 𝐸 ⊆ ran 𝐹
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ran 𝐹)
18		dfss2 3931	. . . 4 ⊢ (𝐸 ⊆ ran 𝐹 ↔ (𝐸 ∩ ran 𝐹) = 𝐸)
19	17, 18	sylib 221	. . 3 ⊢ (𝜑 → (𝐸 ∩ ran 𝐹) = 𝐸)
20	14, 19	eqtrd 2804	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = 𝐸)
21	6, 11, 20	3eqtrd 2808	1 ⊢ (𝜑 → ran 𝑋 = 𝐸)

Syntax hints:   → wi 4   = wceq 1567   ∩ cin 3912   ⊆ wss 3913  ^◡ccnv 5663  ran crn 5665   ↾ cres 5666   “ cima 5667  Fun wfun 6533  ⟶wf 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-fun 6541  df-fn 6542  df-f 6543

This theorem is referenced by:  fcoreslem4  47729  fcoresf1  47732