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Theorem elovmpt3rab1 6935
Description: Implications for the value of an operation defined by the maps-to notation with a function into a class abstraction as a result having an element. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
Hypotheses
Ref Expression
ovmpt3rab1.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
ovmpt3rab1.m ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
ovmpt3rab1.n ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
Assertion
Ref Expression
elovmpt3rab1 ((𝐾𝑈𝐿𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿))))
Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝐿,𝑎,𝑥,𝑦   𝑁,𝑎   𝑥,𝑈,𝑦   𝑋,𝑎,𝑥,𝑦,𝑧   𝑌,𝑎,𝑥,𝑦,𝑧   𝑧,𝐿   𝑧,𝑇   𝑧,𝑈   𝐴,𝑎   𝑍,𝑎,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎)   𝐴(𝑥,𝑦,𝑧)   𝑇(𝑥,𝑦,𝑎)   𝑈(𝑎)   𝐾(𝑎)   𝑀(𝑥,𝑦,𝑧,𝑎)   𝑁(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧,𝑎)   𝑍(𝑥,𝑦)

Proof of Theorem elovmpt3rab1
StepHypRef Expression
1 ovmpt3rab1.o . . . 4 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
21elovmpt3imp 6932 . . 3 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3 simprl 809 . . . . 5 ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4 elfvdm 6258 . . . . . . 7 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → 𝑍 ∈ dom (𝑋𝑂𝑌))
5 simpl 472 . . . . . . . . . . . . . . 15 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
65adantr 480 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑋 ∈ V)
7 simplr 807 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑌 ∈ V)
8 simprl 809 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝐾𝑈)
9 simprr 811 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝐿𝑇)
10 ovmpt3rab1.m . . . . . . . . . . . . . . 15 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
11 ovmpt3rab1.n . . . . . . . . . . . . . . 15 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
121, 10, 11ovmpt3rabdm 6934 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
136, 7, 8, 9, 12syl31anc 1369 . . . . . . . . . . . . 13 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → dom (𝑋𝑂𝑌) = 𝐾)
1413eleq2d 2716 . . . . . . . . . . . 12 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → (𝑍 ∈ dom (𝑋𝑂𝑌) ↔ 𝑍𝐾))
1514biimpcd 239 . . . . . . . . . . 11 (𝑍 ∈ dom (𝑋𝑂𝑌) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑍𝐾))
1615adantr 480 . . . . . . . . . 10 ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑍𝐾))
1716imp 444 . . . . . . . . 9 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → 𝑍𝐾)
18 simpl 472 . . . . . . . . . 10 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝑍𝐾)
19 simplr 807 . . . . . . . . . . . . 13 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍))
2019adantl 481 . . . . . . . . . . . 12 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍))
21 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝐾𝑈𝐿𝑇) → 𝐾𝑈)
2221anim2i 592 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝐾𝑈))
23 df-3an 1056 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝐾𝑈))
2422, 23sylibr 224 . . . . . . . . . . . . . . 15 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈))
2524ad2antll 765 . . . . . . . . . . . . . 14 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈))
26 sbceq1a 3479 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
27 sbceq1a 3479 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
2826, 27sylan9bbr 737 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
29 nfsbc1v 3488 . . . . . . . . . . . . . . . 16 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑦𝑋
31 nfsbc1v 3488 . . . . . . . . . . . . . . . . 17 𝑦[𝑌 / 𝑦]𝜑
3230, 31nfsbc 3490 . . . . . . . . . . . . . . . 16 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
331, 10, 11, 28, 29, 32ovmpt3rab1 6933 . . . . . . . . . . . . . . 15 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
3433fveq1d 6231 . . . . . . . . . . . . . 14 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) → ((𝑋𝑂𝑌)‘𝑍) = ((𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍))
3525, 34syl 17 . . . . . . . . . . . . 13 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → ((𝑋𝑂𝑌)‘𝑍) = ((𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍))
36 rabexg 4844 . . . . . . . . . . . . . . . 16 (𝐿𝑇 → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
3736adantl 481 . . . . . . . . . . . . . . 15 ((𝐾𝑈𝐿𝑇) → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
3837ad2antll 765 . . . . . . . . . . . . . 14 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
39 nfcv 2793 . . . . . . . . . . . . . . 15 𝑧𝑍
40 nfsbc1v 3488 . . . . . . . . . . . . . . . 16 𝑧[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑
41 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑧𝐿
4240, 41nfrab 3153 . . . . . . . . . . . . . . 15 𝑧{𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
43 sbceq1a 3479 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑍 → ([𝑋 / 𝑥][𝑌 / 𝑦]𝜑[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
4443rabbidv 3220 . . . . . . . . . . . . . . 15 (𝑧 = 𝑍 → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} = {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
45 eqid 2651 . . . . . . . . . . . . . . 15 (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
4639, 42, 44, 45fvmptf 6340 . . . . . . . . . . . . . 14 ((𝑍𝐾 ∧ {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V) → ((𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍) = {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
4738, 46sylan2 490 . . . . . . . . . . . . 13 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → ((𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍) = {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
4835, 47eqtr2d 2686 . . . . . . . . . . . 12 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} = ((𝑋𝑂𝑌)‘𝑍))
4920, 48eleqtrrd 2733 . . . . . . . . . . 11 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝐴 ∈ {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
50 elrabi 3391 . . . . . . . . . . 11 (𝐴 ∈ {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝐴𝐿)
5149, 50syl 17 . . . . . . . . . 10 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝐴𝐿)
5218, 51jca 553 . . . . . . . . 9 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → (𝑍𝐾𝐴𝐿))
5317, 52mpancom 704 . . . . . . . 8 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → (𝑍𝐾𝐴𝐿))
5453exp31 629 . . . . . . 7 (𝑍 ∈ dom (𝑋𝑂𝑌) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → (𝑍𝐾𝐴𝐿))))
554, 54mpcom 38 . . . . . 6 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → (𝑍𝐾𝐴𝐿)))
5655imp 444 . . . . 5 ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → (𝑍𝐾𝐴𝐿))
573, 56jca 553 . . . 4 ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿)))
5857exp32 630 . . 3 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((𝐾𝑈𝐿𝑇) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿)))))
592, 58mpd 15 . 2 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝐾𝑈𝐿𝑇) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿))))
6059com12 32 1 ((𝐾𝑈𝐿𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  [wsbc 3468  cmpt 4762  dom cdm 5143  cfv 5926  (class class class)co 6690  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  elovmpt3rab  6936  elovmptnn0wrd  13381
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