Theorem fcoreslem1 47059

Description: Lemma 1 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)

Assertion

Ref	Expression
fcoreslem1	⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))

Proof of Theorem fcoreslem1

Step	Hyp	Ref	Expression
1		fcores.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffund 6715	. . . 4 ⊢ (𝜑 → Fun 𝐹)
3		cnvimainrn 7062	. . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐶)) = (◡𝐹 “ 𝐶))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (◡𝐹 “ (ran 𝐹 ∩ 𝐶)) = (◡𝐹 “ 𝐶))
5	4	eqcomd 2742	. 2 ⊢ (𝜑 → (◡𝐹 “ 𝐶) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶)))
6		fcores.p	. 2 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
7		fcores.e	. . 3 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
8	7	imaeq2i 6050	. 2 ⊢ (◡𝐹 “ 𝐸) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶))
9	5, 6, 8	3eqtr4g 2796	1 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3930  ^◡ccnv 5658  ran crn 5660   “ 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-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:  fcoreslem2  47060