Theorem elovmpod 7613

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 7614 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 11	. . 3 ⊢ (𝜑 → 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
3		elovmpod.c	. . . 4 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷)
4	3	adantl 481	. . 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 7521	. 2 ⊢ (𝜑 → (𝑋𝑂𝑌) = 𝐷)
9	8	eleq2d 2814	1 ⊢ (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  (class class class)co 7369   ∈ cmpo 7371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374

This theorem is referenced by:  isgrim  47855  isgrlim  47954