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Theorem el2mpocl 8072
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 8071 . 2 (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
3 simpl 484 . . . . . . 7 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
4 simplr 768 . . . . . . . 8 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌𝐵)
5 el2mpocl.e . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐶 = 𝐹𝐷 = 𝐺))
65simpld 496 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝐶 = 𝐹)
76adantll 713 . . . . . . . 8 ((((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹)
84, 7csbied 3932 . . . . . . 7 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌 / 𝑦𝐶 = 𝐹)
93, 8csbied 3932 . . . . . 6 ((𝑋𝐴𝑌𝐵) → 𝑋 / 𝑥𝑌 / 𝑦𝐶 = 𝐹)
109eleq2d 2820 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑆𝐹))
115simprd 497 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝐷 = 𝐺)
1211adantll 713 . . . . . . . 8 ((((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
134, 12csbied 3932 . . . . . . 7 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌 / 𝑦𝐷 = 𝐺)
143, 13csbied 3932 . . . . . 6 ((𝑋𝐴𝑌𝐵) → 𝑋 / 𝑥𝑌 / 𝑦𝐷 = 𝐺)
1514eleq2d 2820 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷𝑇𝐺))
1610, 15anbi12d 632 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷) ↔ (𝑆𝐹𝑇𝐺)))
1716biimpd 228 . . 3 ((𝑋𝐴𝑌𝐵) → ((𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷) → (𝑆𝐹𝑇𝐺)))
1817imdistani 570 . 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 397   = wceq 1542  wcel 2107  wral 3062  csb 3894  (class class class)co 7409  cmpo 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  wwlksonvtx  29109  wspthnonp  29113
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