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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 = (𝑎𝐴, 𝑏𝐵𝐶)
elovmpt2.c 𝐶 ∈ V
elovmpt2.e ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)
Assertion
Ref Expression
elovmpt2 (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
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
StepHypRef Expression
1 elovmpt2.d . . . 4 𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)
21elmpt2cl 7041 . . 3 (𝐹 ∈ (𝑋𝐷𝑌) → (𝑋𝐴𝑌𝐵))
3 elovmpt2.c . . . . . . 7 𝐶 ∈ V
43gen2 1872 . . . . . 6 𝑎𝑏 𝐶 ∈ V
5 elovmpt2.e . . . . . . . 8 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)
65eleq1d 2824 . . . . . . 7 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝐶 ∈ V ↔ 𝐸 ∈ V))
76spc2gv 3436 . . . . . 6 ((𝑋𝐴𝑌𝐵) → (∀𝑎𝑏 𝐶 ∈ V → 𝐸 ∈ V))
84, 7mpi 20 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝐸 ∈ V)
95, 1ovmpt2ga 6955 . . . . 5 ((𝑋𝐴𝑌𝐵𝐸 ∈ V) → (𝑋𝐷𝑌) = 𝐸)
108, 9mpd3an3 1574 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝑋𝐷𝑌) = 𝐸)
1110eleq2d 2825 . . 3 ((𝑋𝐴𝑌𝐵) → (𝐹 ∈ (𝑋𝐷𝑌) ↔ 𝐹𝐸))
122, 11biadan2 677 . 2 (𝐹 ∈ (𝑋𝐷𝑌) ↔ ((𝑋𝐴𝑌𝐵) ∧ 𝐹𝐸))
13 df-3an 1074 . 2 ((𝑋𝐴𝑌𝐵𝐹𝐸) ↔ ((𝑋𝐴𝑌𝐵) ∧ 𝐹𝐸))
1412, 13bitr4i 267 1 (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
Colors of variables: wff setvar class
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
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