Theorem elovmpt2 7044

Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 17861, islmhm 19229. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses

Ref	Expression
elovmpt2.d	⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
elovmpt2.c	⊢ 𝐶 ∈ V
elovmpt2.e	⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸)

Assertion

Ref	Expression
elovmpt2	⊢ (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 elovmpt2

Step	Hyp	Ref	Expression
1		elovmpt2.d	. . . 4 ⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
2	1	elmpt2cl 7041	. . 3 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
3		elovmpt2.c	. . . . . . 7 ⊢ 𝐶 ∈ V
4	3	gen2 1872	. . . . . 6 ⊢ ∀𝑎∀𝑏 𝐶 ∈ V
5		elovmpt2.e	. . . . . . . 8 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸)
6	5	eleq1d 2824	. . . . . . 7 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝐶 ∈ V ↔ 𝐸 ∈ V))
7	6	spc2gv 3436	. . . . . 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	ovmpt2ga 6955	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐸 ∈ V) → (𝑋𝐷𝑌) = 𝐸)
10	8, 9	mpd3an3 1574	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐷𝑌) = 𝐸)
11	10	eleq2d 2825	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋𝐷𝑌) ↔ 𝐹 ∈ 𝐸))
12	2, 11	biadan2 677	. 2 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ 𝐸))
13		df-3an 1074	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ 𝐸))
14	12, 13	bitr4i 267	1 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072  ∀wal 1630   = wceq 1632   ∈ wcel 2139  Vcvv 3340  (class class class)co 6813   ↦ cmpt2 6815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818

This theorem is referenced by:  isgim  17905  oppglsm  18257  islmim  19264