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Theorem ovmpt2dxf 5752
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 5651 . 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 2088 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 5749 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 9 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 1460 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 2850 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 1460 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 2850 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 270 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐿)
15 simplr 497 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
163ad2antrr 472 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐶)
1715, 16eqeltrd 2164 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
185ad2antrr 472 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐿)
19 simpr 108 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20 ovmpt2dx.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
2120adantr 270 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐷 = 𝐿)
2218, 19, 213eltr4d 2171 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
23 ovmpt2dx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2423anassrs 392 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
25 ovmpt2dx.6 . . . . . . . . . 10 (𝜑𝑆𝑋)
26 elex 2630 . . . . . . . . . 10 (𝑆𝑋𝑆 ∈ V)
2725, 26syl 14 . . . . . . . . 9 (𝜑𝑆 ∈ V)
2827ad2antrr 472 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ V)
2924, 28eqeltrd 2164 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ V)
30 biimt 239 . . . . . . 7 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3117, 22, 29, 30syl3anc 1174 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3215, 19oveq12d 5652 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3332, 24eqeq12d 2102 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3431, 33bitr3d 188 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
35 ovmpt2dxf.ay . . . . . . 7 𝑦𝐴
3635nfeq2 2240 . . . . . 6 𝑦 𝑥 = 𝐴
376, 36nfan 1502 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
38 nfmpt22 5698 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
39 nfcv 2228 . . . . . . . 8 𝑦𝐵
4035, 38, 39nfov 5661 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
41 ovmpt2dxf.sy . . . . . . 7 𝑦𝑆
4240, 41nfeq 2236 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4342a1i 9 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4414, 34, 37, 43sbciedf 2872 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
45 nfcv 2228 . . . . . . 7 𝑥𝐴
46 nfmpt21 5697 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
47 ovmpt2dxf.bx . . . . . . 7 𝑥𝐵
4845, 46, 47nfov 5661 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
49 ovmpt2dxf.sx . . . . . 6 𝑥𝑆
5048, 49nfeq 2236 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5150a1i 9 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
523, 44, 4, 51sbciedf 2872 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5313, 52mpbid 145 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
542, 53eqtrd 2120 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215  Vcvv 2619  [wsbc 2838  (class class class)co 5634  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639
This theorem is referenced by:  ovmpt2dx  5753  mpt2xopoveq  5987
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