Theorem fcoreslem1 47452

Description: Lemma 1 for fcores 47456. (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 6676	. . . 4 ⊢ (𝜑 → Fun 𝐹)
3		cnvimainrn 7023	. . . 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 2743	. 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 6027	. 2 ⊢ (◡𝐹 “ 𝐸) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶))
9	5, 6, 8	3eqtr4g 2797	1 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3902  ^◡ccnv 5633  ran crn 5635   “ cima 5637  Fun wfun 6496  ⟶wf 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6504  df-fn 6505  df-f 6506

This theorem is referenced by:  fcoreslem2  47453