Theorem fcoreslem1 47346

Description: Lemma 1 for fcores 47350. (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 6665	. . . 4 ⊢ (𝜑 → Fun 𝐹)
3		cnvimainrn 7012	. . . 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 2741	. 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 6016	. 2 ⊢ (◡𝐹 “ 𝐸) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶))
9	5, 6, 8	3eqtr4g 2795	1 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3899  ^◡ccnv 5622  ran crn 5624   “ cima 5626  Fun wfun 6485  ⟶wf 6487

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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 2538  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6493  df-fn 6494  df-f 6495

This theorem is referenced by:  fcoreslem2  47347