Theorem fcoreslem2 46979

Description: Lemma 2 for fcores 46982. (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 5713	. . 3 ⊢ (𝐹 “ 𝑃) = ran (𝐹 ↾ 𝑃)
2		fcores.x	. . . . . 6 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
3	2	rneqi 5962	. . . . 5 ⊢ ran 𝑋 = ran (𝐹 ↾ 𝑃)
4	3	eqcomi 2749	. . . 4 ⊢ ran (𝐹 ↾ 𝑃) = ran 𝑋
5	4	a1i 11	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝑃) = ran 𝑋)
6	1, 5	eqtr2id 2793	. 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 46978	. . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
11	10	imaeq2d 6089	. 2 ⊢ (𝜑 → (𝐹 “ 𝑃) = (𝐹 “ (◡𝐹 “ 𝐸)))
12	7	ffund 6751	. . . 4 ⊢ (𝜑 → Fun 𝐹)
13		funimacnv 6659	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹))
15		inss1 4258	. . . . . 6 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ ran 𝐹
16	8, 15	eqsstri 4043	. . . . 5 ⊢ 𝐸 ⊆ ran 𝐹
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ran 𝐹)
18		dfss2 3994	. . . 4 ⊢ (𝐸 ⊆ ran 𝐹 ↔ (𝐸 ∩ ran 𝐹) = 𝐸)
19	17, 18	sylib 218	. . 3 ⊢ (𝜑 → (𝐸 ∩ ran 𝐹) = 𝐸)
20	14, 19	eqtrd 2780	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = 𝐸)
21	6, 11, 20	3eqtrd 2784	1 ⊢ (𝜑 → ran 𝑋 = 𝐸)

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3975   ⊆ wss 3976  ^◡ccnv 5699  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  fcoreslem4  46981  fcoresf1  46984