Theorem fcoreslem1 44597

Description: Lemma 1 for fcores 44601. (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 6622	. . . 4 ⊢ (𝜑 → Fun 𝐹)
3		cnvimainrn 6964	. . . 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 2739	. 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 5968	. 2 ⊢ (◡𝐹 “ 𝐸) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶))
9	5, 6, 8	3eqtr4g 2798	1 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3888  ^◡ccnv 5590  ran crn 5592   “ cima 5594  Fun wfun 6441  ⟶wf 6443

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 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-br 5078  df-opab 5140  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-fun 6449  df-fn 6450  df-f 6451

This theorem is referenced by:  fcoreslem2  44598