ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovmpt2dxf GIF version

Theorem ovmpt2dxf 5653
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 5556 . 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 2056 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 5650 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 9 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 1431 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 2798 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 1431 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 2798 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 265 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐿)
15 simplr 490 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
163ad2antrr 465 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐶)
1715, 16eqeltrd 2130 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
185ad2antrr 465 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐿)
19 simpr 107 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20 ovmpt2dx.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
2120adantr 265 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐷 = 𝐿)
2218, 19, 213eltr4d 2137 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
23 ovmpt2dx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2423anassrs 386 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
25 ovmpt2dx.6 . . . . . . . . . 10 (𝜑𝑆𝑋)
26 elex 2583 . . . . . . . . . 10 (𝑆𝑋𝑆 ∈ V)
2725, 26syl 14 . . . . . . . . 9 (𝜑𝑆 ∈ V)
2827ad2antrr 465 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ V)
2924, 28eqeltrd 2130 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ V)
30 biimt 234 . . . . . . 7 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3117, 22, 29, 30syl3anc 1146 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3215, 19oveq12d 5557 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3332, 24eqeq12d 2070 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3431, 33bitr3d 183 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
35 ovmpt2dxf.ay . . . . . . 7 𝑦𝐴
3635nfeq2 2205 . . . . . 6 𝑦 𝑥 = 𝐴
376, 36nfan 1473 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
38 nfmpt22 5599 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
39 nfcv 2194 . . . . . . . 8 𝑦𝐵
4035, 38, 39nfov 5562 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
41 ovmpt2dxf.sy . . . . . . 7 𝑦𝑆
4240, 41nfeq 2201 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4342a1i 9 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4414, 34, 37, 43sbciedf 2820 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
45 nfcv 2194 . . . . . . 7 𝑥𝐴
46 nfmpt21 5598 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
47 ovmpt2dxf.bx . . . . . . 7 𝑥𝐵
4845, 46, 47nfov 5562 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
49 ovmpt2dxf.sx . . . . . 6 𝑥𝑆
5048, 49nfeq 2201 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5150a1i 9 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
523, 44, 4, 51sbciedf 2820 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5313, 52mpbid 139 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
542, 53eqtrd 2088 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181  Vcvv 2574  [wsbc 2786  (class class class)co 5539  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544
This theorem is referenced by:  ovmpt2dx  5654  mpt2xopoveq  5885
  Copyright terms: Public domain W3C validator