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Theorem el2mpt2cl 7401
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
el2mpt2cl.o 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
el2mpt2cl.e ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐶 = 𝐹𝐷 = 𝐺))
Assertion
Ref Expression
el2mpt2cl (∀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 el2mpt2cl
StepHypRef Expression
1 el2mpt2cl.o . . 3 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
21el2mpt2csbcl 7400 . 2 (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
3 simpl 468 . . . . . . 7 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
4 simplr 752 . . . . . . . 8 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌𝐵)
5 el2mpt2cl.e . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐶 = 𝐹𝐷 = 𝐺))
65simpld 482 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝐶 = 𝐹)
76adantll 693 . . . . . . . 8 ((((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹)
84, 7csbied 3709 . . . . . . 7 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌 / 𝑦𝐶 = 𝐹)
93, 8csbied 3709 . . . . . 6 ((𝑋𝐴𝑌𝐵) → 𝑋 / 𝑥𝑌 / 𝑦𝐶 = 𝐹)
109eleq2d 2836 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑆𝐹))
115simprd 483 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝐷 = 𝐺)
1211adantll 693 . . . . . . . 8 ((((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
134, 12csbied 3709 . . . . . . 7 (((𝑋𝐴𝑌𝐵) ∧ 𝑥 = 𝑋) → 𝑌 / 𝑦𝐷 = 𝐺)
143, 13csbied 3709 . . . . . 6 ((𝑋𝐴𝑌𝐵) → 𝑋 / 𝑥𝑌 / 𝑦𝐷 = 𝐺)
1514eleq2d 2836 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷𝑇𝐺))
1610, 15anbi12d 616 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷) ↔ (𝑆𝐹𝑇𝐺)))
1716biimpd 219 . . 3 ((𝑋𝐴𝑌𝐵) → ((𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷) → (𝑆𝐹𝑇𝐺)))
1817imdistani 558 . 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 382   = wceq 1631  wcel 2145  wral 3061  csb 3682  (class class class)co 6793  cmpt2 6795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316
This theorem is referenced by:  wwlksonvtx  26985  wspthnonp  26993
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