Theorem elovmpod 7614

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 7615 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 7522	. 2 ⊢ (𝜑 → (𝑋𝑂𝑌) = 𝐷)
9	8	eleq2d 2823	1 ⊢ (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  (class class class)co 7370   ∈ cmpo 7372

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-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375

This theorem is referenced by:  isgrim  48271  isgrlim  48371