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Theorem elovmpo 7375
Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 18349, islmhm 19790. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses
Ref Expression
elovmpo.d 𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)
elovmpo.c 𝐶 ∈ V
elovmpo.e ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)
Assertion
Ref Expression
elovmpo (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 elovmpo
StepHypRef Expression
1 elovmpo.d . . . 4 𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)
21elmpocl 7372 . . 3 (𝐹 ∈ (𝑋𝐷𝑌) → (𝑋𝐴𝑌𝐵))
3 elovmpo.c . . . . . . 7 𝐶 ∈ V
43gen2 1798 . . . . . 6 𝑎𝑏 𝐶 ∈ V
5 elovmpo.e . . . . . . . 8 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)
65eleq1d 2898 . . . . . . 7 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝐶 ∈ V ↔ 𝐸 ∈ V))
76spc2gv 3576 . . . . . 6 ((𝑋𝐴𝑌𝐵) → (∀𝑎𝑏 𝐶 ∈ V → 𝐸 ∈ V))
84, 7mpi 20 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝐸 ∈ V)
95, 1ovmpoga 7288 . . . . 5 ((𝑋𝐴𝑌𝐵𝐸 ∈ V) → (𝑋𝐷𝑌) = 𝐸)
108, 9mpd3an3 1459 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝑋𝐷𝑌) = 𝐸)
1110eleq2d 2899 . . 3 ((𝑋𝐴𝑌𝐵) → (𝐹 ∈ (𝑋𝐷𝑌) ↔ 𝐹𝐸))
122, 11biadanii 821 . 2 (𝐹 ∈ (𝑋𝐷𝑌) ↔ ((𝑋𝐴𝑌𝐵) ∧ 𝐹𝐸))
13 df-3an 1086 . 2 ((𝑋𝐴𝑌𝐵𝐹𝐸) ↔ ((𝑋𝐴𝑌𝐵) ∧ 𝐹𝐸))
1412, 13bitr4i 281 1 (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2114  Vcvv 3469  (class class class)co 7140  cmpo 7142
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145
This theorem is referenced by:  isgim  18393  oppglsm  18758  islmim  19825
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