Theorem fcoreslem2 47252

Description: Lemma 2 for fcores 47255. (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 5635	. . 3 ⊢ (𝐹 “ 𝑃) = ran (𝐹 ↾ 𝑃)
2		fcores.x	. . . . . 6 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
3	2	rneqi 5884	. . . . 5 ⊢ ran 𝑋 = ran (𝐹 ↾ 𝑃)
4	3	eqcomi 2743	. . . 4 ⊢ ran (𝐹 ↾ 𝑃) = ran 𝑋
5	4	a1i 11	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝑃) = ran 𝑋)
6	1, 5	eqtr2id 2782	. 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 47251	. . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
11	10	imaeq2d 6017	. 2 ⊢ (𝜑 → (𝐹 “ 𝑃) = (𝐹 “ (◡𝐹 “ 𝐸)))
12	7	ffund 6664	. . . 4 ⊢ (𝜑 → Fun 𝐹)
13		funimacnv 6571	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
15		inss1 4187	. . . . . 6 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ ran 𝐹
16	8, 15	eqsstri 3978	. . . . 5 ⊢ 𝐸 ⊆ ran 𝐹
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ran 𝐹)
18		dfss2 3917	. . . 4 ⊢ (𝐸 ⊆ ran 𝐹 ↔ (𝐸 ∩ ran 𝐹) = 𝐸)
19	17, 18	sylib 218	. . 3 ⊢ (𝜑 → (𝐸 ∩ ran 𝐹) = 𝐸)
20	14, 19	eqtrd 2769	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = 𝐸)
21	6, 11, 20	3eqtrd 2773	1 ⊢ (𝜑 → ran 𝑋 = 𝐸)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3898   ⊆ wss 3899  ^◡ccnv 5621  ran crn 5623   ↾ cres 5624   “ cima 5625  Fun wfun 6484  ⟶wf 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-fun 6492  df-fn 6493  df-f 6494

This theorem is referenced by:  fcoreslem4  47254  fcoresf1  47257