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Theorem elovmpt3rab1 7389
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 7386 . . 3 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3 simprl 770 . . . . 5 ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4 elfvdm 6681 . . . . . . 7 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → 𝑍 ∈ dom (𝑋𝑂𝑌))
5 simpl 486 . . . . . . . . . . . . . . 15 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
65adantr 484 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑋 ∈ V)
7 simplr 768 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑌 ∈ V)
8 simprl 770 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝐾𝑈)
9 simprr 772 . . . . . . . . . . . . . 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 7388 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
136, 7, 8, 9, 12syl31anc 1370 . . . . . . . . . . . . 13 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → dom (𝑋𝑂𝑌) = 𝐾)
1413eleq2d 2878 . . . . . . . . . . . 12 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → (𝑍 ∈ dom (𝑋𝑂𝑌) ↔ 𝑍𝐾))
1514biimpcd 252 . . . . . . . . . . 11 (𝑍 ∈ dom (𝑋𝑂𝑌) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑍𝐾))
1615adantr 484 . . . . . . . . . 10 ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → 𝑍𝐾))
1716imp 410 . . . . . . . . 9 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → 𝑍𝐾)
18 simpl 486 . . . . . . . . . 10 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝑍𝐾)
19 simplr 768 . . . . . . . . . . . . 13 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍))
2019adantl 485 . . . . . . . . . . . 12 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍))
21 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝐾𝑈𝐿𝑇) → 𝐾𝑈)
2221anim2i 619 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝐾𝑈))
23 df-3an 1086 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝐾𝑈))
2422, 23sylibr 237 . . . . . . . . . . . . . . 15 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈))
2524ad2antll 728 . . . . . . . . . . . . . 14 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈))
26 sbceq1a 3734 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
27 sbceq1a 3734 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
2826, 27sylan9bbr 514 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
29 nfsbc1v 3743 . . . . . . . . . . . . . . . 16 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30 nfcv 2958 . . . . . . . . . . . . . . . . 17 𝑦𝑋
31 nfsbc1v 3743 . . . . . . . . . . . . . . . . 17 𝑦[𝑌 / 𝑦]𝜑
3230, 31nfsbcw 3745 . . . . . . . . . . . . . . . 16 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
331, 10, 11, 28, 29, 32ovmpt3rab1 7387 . . . . . . . . . . . . . . 15 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
3433fveq1d 6651 . . . . . . . . . . . . . 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 5201 . . . . . . . . . . . . . . . 16 (𝐿𝑇 → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
3736adantl 485 . . . . . . . . . . . . . . 15 ((𝐾𝑈𝐿𝑇) → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
3837ad2antll 728 . . . . . . . . . . . . . 14 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
39 nfcv 2958 . . . . . . . . . . . . . . 15 𝑧𝑍
40 nfsbc1v 3743 . . . . . . . . . . . . . . . 16 𝑧[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑
41 nfcv 2958 . . . . . . . . . . . . . . . 16 𝑧𝐿
4240, 41nfrabw 3341 . . . . . . . . . . . . . . 15 𝑧{𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
43 sbceq1a 3734 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑍 → ([𝑋 / 𝑥][𝑌 / 𝑦]𝜑[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
4443rabbidv 3430 . . . . . . . . . . . . . . 15 (𝑧 = 𝑍 → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} = {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
45 eqid 2801 . . . . . . . . . . . . . . 15 (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
4639, 42, 44, 45fvmptf 6770 . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . 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 2837 . . . . . . . . . . . 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 2896 . . . . . . . . . . 11 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → 𝐴 ∈ {𝑎𝐿[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
50 elrabi 3626 . . . . . . . . . . 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 515 . . . . . . . . 9 ((𝑍𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇)))) → (𝑍𝐾𝐴𝐿))
5317, 52mpancom 687 . . . . . . . 8 (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → (𝑍𝐾𝐴𝐿))
5453exp31 423 . . . . . . 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 410 . . . . 5 ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → (𝑍𝐾𝐴𝐿))
573, 56jca 515 . . . 4 ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾𝑈𝐿𝑇))) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿)))
5857exp32 424 . . 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 399  w3a 1084   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  [wsbc 3723  cmpt 5113  dom cdm 5523  cfv 6328  (class class class)co 7139  cmpo 7141
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144
This theorem is referenced by:  elovmpt3rab  7390  elovmptnn0wrd  13906
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