Theorem elovmpo 7585

Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify islmhm 20915. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses

Ref	Expression
elovmpo.d	⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
elovmpo.c	⊢ 𝐶 ∈ V
elovmpo.e	⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸)

Assertion

Ref	Expression
elovmpo	⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝐸,𝑎,𝑏   𝐹,𝑎,𝑏   𝑋,𝑎,𝑏   𝑌,𝑎,𝑏

Allowed substitution hints:   𝐶(𝑎,𝑏)   𝐷(𝑎,𝑏)

Proof of Theorem elovmpo

Step	Hyp	Ref	Expression
1		elovmpo.d	. . . 4 ⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
2	1	elmpocl 7581	. . 3 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
3		elovmpo.c	. . . . . . 7 ⊢ 𝐶 ∈ V
4	3	gen2 1796	. . . . . 6 ⊢ ∀𝑎∀𝑏 𝐶 ∈ V
5		elovmpo.e	. . . . . . . 8 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸)
6	5	eleq1d 2813	. . . . . . 7 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝐶 ∈ V ↔ 𝐸 ∈ V))
7	6	spc2gv 3552	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑎∀𝑏 𝐶 ∈ V → 𝐸 ∈ V))
8	4, 7	mpi 20	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝐸 ∈ V)
9	5, 1	ovmpoga 7494	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐸 ∈ V) → (𝑋𝐷𝑌) = 𝐸)
10	8, 9	mpd3an3 1464	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐷𝑌) = 𝐸)
11	10	eleq2d 2814	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋𝐷𝑌) ↔ 𝐹 ∈ 𝐸))
12	2, 11	biadanii 821	. 2 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ 𝐸))
13		df-3an 1088	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ 𝐸))
14	12, 13	bitr4i 278	1 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3433  (class class class)co 7340   ∈ cmpo 7342

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 5231  ax-nul 5241  ax-pr 5367

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 3393  df-v 3435  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345

This theorem is referenced by:  isgim  19128  oppglsm  19508  islmim  20950  sn-isghm  42663