Theorem elovmpod 6116

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6117 in deduction form. (Revised by AV, 20-Apr-2025.)

Hypotheses

Ref	Expression
elovmpod.o	⊢ 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
elovmpod.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
elovmpod.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
elovmpod.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
elovmpod.c	⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷)

Assertion

Ref	Expression
elovmpod	⊢ (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))

Distinct variable groups:   𝐷,𝑎,𝑏   𝑋,𝑎,𝑏   𝑌,𝑎,𝑏   𝜑,𝑎,𝑏

Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝐸(𝑎,𝑏)   𝑂(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem elovmpod

Step	Hyp	Ref	Expression
1		elovmpod.o	. . . 4 ⊢ 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
2	1	a1i 9	. . 3 ⊢ (𝜑 → 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
3		elovmpod.c	. . . 4 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷)
4	3	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝑋 ∧ 𝑏 = 𝑌)) → 𝐶 = 𝐷)
5		elovmpod.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
6		elovmpod.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		elovmpod.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
8	2, 4, 5, 6, 7	ovmpod 6046	. 2 ⊢ (𝜑 → (𝑋𝑂𝑌) = 𝐷)
9	8	eleq2d 2263	1 ⊢ (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  (class class class)co 5918   ∈ cmpo 5920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923

This theorem is referenced by: (None)