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Theorem el2mpocl 7768
Description: If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. Using implicit substitution. (Contributed by AV, 21-May-2021.)
Hypotheses
Ref Expression
el2mpocl.o 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
el2mpocl.e ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐶 = 𝐹𝐷 = 𝐺))
Assertion
Ref Expression
el2mpocl (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐹𝑇𝐺))))
Distinct variable groups:   𝐴,𝑠,𝑡,𝑥,𝑦   𝐵,𝑠,𝑡,𝑥,𝑦   𝐶,𝑠,𝑡   𝐷,𝑠,𝑡   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑋,𝑠,𝑡,𝑥,𝑦   𝑌,𝑠,𝑡,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑡,𝑠)   𝑇(𝑥,𝑦,𝑡,𝑠)   𝑈(𝑡,𝑠)   𝐸(𝑥,𝑦,𝑡,𝑠)   𝐹(𝑡,𝑠)   𝐺(𝑡,𝑠)   𝑂(𝑥,𝑦,𝑡,𝑠)   𝑉(𝑡,𝑠)   𝑊(𝑥,𝑦,𝑡,𝑠)

Proof of Theorem el2mpocl
StepHypRef Expression
1 el2mpocl.o . . 3 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
21el2mpocsbcl 7767 . 2 (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
3 simpl 486 . . . . . . 7 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
4 simplr 768 . . . . . . . 8 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌𝐵)
5 el2mpocl.e . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐶 = 𝐹𝐷 = 𝐺))
65simpld 498 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝐶 = 𝐹)
76adantll 713 . . . . . . . 8 ((((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹)
84, 7csbied 3891 . . . . . . 7 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌 / 𝑦𝐶 = 𝐹)
93, 8csbied 3891 . . . . . 6 ((𝑋𝐴𝑌𝐵) → 𝑋 / 𝑥𝑌 / 𝑦𝐶 = 𝐹)
109eleq2d 2899 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑆𝐹))
115simprd 499 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝐷 = 𝐺)
1211adantll 713 . . . . . . . 8 ((((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
134, 12csbied 3891 . . . . . . 7 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌 / 𝑦𝐷 = 𝐺)
143, 13csbied 3891 . . . . . 6 ((𝑋𝐴𝑌𝐵) → 𝑋 / 𝑥𝑌 / 𝑦𝐷 = 𝐺)
1514eleq2d 2899 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷𝑇𝐺))
1610, 15anbi12d 633 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷) ↔ (𝑆𝐹𝑇𝐺)))
1716biimpd 232 . . 3 ((𝑋𝐴𝑌𝐵) → ((𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷) → (𝑆𝐹𝑇𝐺)))
1817imdistani 572 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐹𝑇𝐺)))
192, 18syl6 35 1 (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐹𝑇𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  csb 3855  (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-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  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-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676
This theorem is referenced by:  wwlksonvtx  27639  wspthnonp  27643
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