Theorem fcoreslem1 44444

Description: Lemma 1 for fcores 44448. (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 6588	. . . 4 ⊢ (𝜑 → Fun 𝐹)
3		cnvimainrn 6926	. . . 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 2744	. 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 5956	. 2 ⊢ (◡𝐹 “ 𝐸) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶))
9	5, 6, 8	3eqtr4g 2804	1 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))

Syntax hints:   → wi 4   = wceq 1539   ∩ cin 3882  ^◡ccnv 5579  ran crn 5581   “ cima 5583  Fun wfun 6412  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-fun 6420  df-fn 6421  df-f 6422

This theorem is referenced by:  fcoreslem2  44445