Theorem fcoreslem2 48050

Description: Lemma 2 for fcores 48053. (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 5661	. . 3 ⊢ (𝐹 “ 𝑃) = ran (𝐹 ↾ 𝑃)
2		fcores.x	. . . . . 6 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
3	2	rneqi 5916	. . . . 5 ⊢ ran 𝑋 = ran (𝐹 ↾ 𝑃)
4	3	eqcomi 2769	. . . 4 ⊢ ran (𝐹 ↾ 𝑃) = ran 𝑋
5	4	a1i 11	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝑃) = ran 𝑋)
6	1, 5	eqtr2id 2808	. 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 48049	. . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
11	10	imaeq2d 6051	. 2 ⊢ (𝜑 → (𝐹 “ 𝑃) = (𝐹 “ (◡𝐹 “ 𝐸)))
12	7	ffund 6703	. . . 4 ⊢ (𝜑 → Fun 𝐹)
13		funimacnv 6610	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
15		inss1 4182	. . . . . 6 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ ran 𝐹
16	8, 15	eqsstri 3977	. . . . 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 221	. . 3 ⊢ (𝜑 → (𝐸 ∩ ran 𝐹) = 𝐸)
20	14, 19	eqtrd 2795	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = 𝐸)
21	6, 11, 20	3eqtrd 2799	1 ⊢ (𝜑 → ran 𝑋 = 𝐸)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 3898   ⊆ wss 3899  ^◡ccnv 5647  ran crn 5649   ↾ cres 5650   “ cima 5651  Fun wfun 6522  ⟶wf 6524

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-fun 6530  df-fn 6531  df-f 6532

This theorem is used by:  fcoreslem4  48052  fcoresf1  48055