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Theorem ovmpt2dxf 6828
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2dx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpt2dx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2dx.3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
ovmpt2dx.4 (𝜑𝐴𝐶)
ovmpt2dx.5 (𝜑𝐵𝐿)
ovmpt2dx.6 (𝜑𝑆𝑋)
ovmpt2dxf.px 𝑥𝜑
ovmpt2dxf.py 𝑦𝜑
ovmpt2dxf.ay 𝑦𝐴
ovmpt2dxf.bx 𝑥𝐵
ovmpt2dxf.sx 𝑥𝑆
ovmpt2dxf.sy 𝑦𝑆
Assertion
Ref Expression
ovmpt2dxf (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpt2dxf
StepHypRef Expression
1 ovmpt2dx.1 . . 3 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
21oveqd 6707 . 2 (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3 ovmpt2dx.4 . . . 4 (𝜑𝐴𝐶)
4 ovmpt2dxf.px . . . . 5 𝑥𝜑
5 ovmpt2dx.5 . . . . . 6 (𝜑𝐵𝐿)
6 ovmpt2dxf.py . . . . . . 7 𝑦𝜑
7 eqid 2651 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 6825 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 11 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 2120 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 3482 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 2120 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 3482 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 480 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐿)
15 simplr 807 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
163ad2antrr 762 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐶)
1715, 16eqeltrd 2730 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
185ad2antrr 762 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐿)
19 simpr 476 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20 ovmpt2dx.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
2120adantr 480 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐷 = 𝐿)
2218, 19, 213eltr4d 2745 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
23 ovmpt2dx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2423anassrs 681 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
25 ovmpt2dx.6 . . . . . . . . . 10 (𝜑𝑆𝑋)
26 elex 3243 . . . . . . . . . 10 (𝑆𝑋𝑆 ∈ V)
2725, 26syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ V)
2827ad2antrr 762 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ V)
2924, 28eqeltrd 2730 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ V)
30 biimt 349 . . . . . . 7 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3117, 22, 29, 30syl3anc 1366 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3215, 19oveq12d 6708 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3332, 24eqeq12d 2666 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3431, 33bitr3d 270 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
35 ovmpt2dxf.ay . . . . . . 7 𝑦𝐴
3635nfeq2 2809 . . . . . 6 𝑦 𝑥 = 𝐴
376, 36nfan 1868 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
38 nfmpt22 6765 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
39 nfcv 2793 . . . . . . . 8 𝑦𝐵
4035, 38, 39nfov 6716 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
41 ovmpt2dxf.sy . . . . . . 7 𝑦𝑆
4240, 41nfeq 2805 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4342a1i 11 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4414, 34, 37, 43sbciedf 3504 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
45 nfcv 2793 . . . . . . 7 𝑥𝐴
46 nfmpt21 6764 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
47 ovmpt2dxf.bx . . . . . . 7 𝑥𝐵
4845, 46, 47nfov 6716 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
49 ovmpt2dxf.sx . . . . . 6 𝑥𝑆
5048, 49nfeq 2805 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5150a1i 11 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
523, 44, 4, 51sbciedf 3504 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5313, 52mpbid 222 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
542, 53eqtrd 2685 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780  Vcvv 3231  [wsbc 3468  (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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  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-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  ovmpt2dx  6829  elovmpt2rab  6922  elovmpt2rab1  6923  ovmpt3rab1  6933  mpt2xopoveq  7390  fvmpt2curryd  7442  mdetralt2  20463  mdetunilem2  20467  gsummatr01lem4  20512
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