Theorem opf2fval 49895

Description: The morphism part of the op functor on functor categories. Lemma for fucoppc 49900. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
opf2fval.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))
opf2fval.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
opf2fval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
opf2fval	⊢ (𝜑 → (𝑋𝐹𝑌) = ( I ↾ (𝑌𝑁𝑋)))

Distinct variable groups:   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem opf2fval

Step	Hyp	Ref	Expression
1		opf2fval.f	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))
2		simprr 778	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
3		simprl 776	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
4	2, 3	oveq12d 7374	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝑁𝑥) = (𝑌𝑁𝑋))
5	4	reseq2d 5931	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ (𝑦𝑁𝑥)) = ( I ↾ (𝑌𝑁𝑋)))
6		opf2fval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
7		opf2fval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		ovexd 7391	. . 3 ⊢ (𝜑 → (𝑌𝑁𝑋) ∈ V)
9		resiexg 7852	. . 3 ⊢ ((𝑌𝑁𝑋) ∈ V → ( I ↾ (𝑌𝑁𝑋)) ∈ V)
10	8, 9	syl 17	. 2 ⊢ (𝜑 → ( I ↾ (𝑌𝑁𝑋)) ∈ V)
11	1, 5, 6, 7, 10	ovmpod 7508	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = ( I ↾ (𝑌𝑁𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   I cid 5512   ↾ cres 5620  (class class class)co 7356   ∈ cmpo 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361

This theorem is referenced by:  opf2  49896  fucoppc  49900