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Theorem ovmpodxf 5903
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpodx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpodx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpodx.3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
ovmpodx.4 (𝜑𝐴𝐶)
ovmpodx.5 (𝜑𝐵𝐿)
ovmpodx.6 (𝜑𝑆𝑋)
ovmpodxf.px 𝑥𝜑
ovmpodxf.py 𝑦𝜑
ovmpodxf.ay 𝑦𝐴
ovmpodxf.bx 𝑥𝐵
ovmpodxf.sx 𝑥𝑆
ovmpodxf.sy 𝑦𝑆
Assertion
Ref Expression
ovmpodxf (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 ovmpodxf
StepHypRef Expression
1 ovmpodx.1 . . 3 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
21oveqd 5798 . 2 (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3 ovmpodx.4 . . . 4 (𝜑𝐴𝐶)
4 ovmpodxf.px . . . . 5 𝑥𝜑
5 ovmpodx.5 . . . . . 6 (𝜑𝐵𝐿)
6 ovmpodxf.py . . . . . . 7 𝑦𝜑
7 eqid 2140 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 5900 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 9 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 1503 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 2924 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 1503 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 2924 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 274 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐿)
15 simplr 520 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
163ad2antrr 480 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐶)
1715, 16eqeltrd 2217 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
185ad2antrr 480 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐿)
19 simpr 109 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20 ovmpodx.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
2120adantr 274 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐷 = 𝐿)
2218, 19, 213eltr4d 2224 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
23 ovmpodx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2423anassrs 398 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
25 ovmpodx.6 . . . . . . . . . 10 (𝜑𝑆𝑋)
26 elex 2700 . . . . . . . . . 10 (𝑆𝑋𝑆 ∈ V)
2725, 26syl 14 . . . . . . . . 9 (𝜑𝑆 ∈ V)
2827ad2antrr 480 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ V)
2924, 28eqeltrd 2217 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ V)
30 biimt 240 . . . . . . 7 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3117, 22, 29, 30syl3anc 1217 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3215, 19oveq12d 5799 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3332, 24eqeq12d 2155 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3431, 33bitr3d 189 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
35 ovmpodxf.ay . . . . . . 7 𝑦𝐴
3635nfeq2 2294 . . . . . 6 𝑦 𝑥 = 𝐴
376, 36nfan 1545 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
38 nfmpo2 5846 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
39 nfcv 2282 . . . . . . . 8 𝑦𝐵
4035, 38, 39nfov 5808 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
41 ovmpodxf.sy . . . . . . 7 𝑦𝑆
4240, 41nfeq 2290 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4342a1i 9 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4414, 34, 37, 43sbciedf 2947 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
45 nfcv 2282 . . . . . . 7 𝑥𝐴
46 nfmpo1 5845 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
47 ovmpodxf.bx . . . . . . 7 𝑥𝐵
4845, 46, 47nfov 5808 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
49 ovmpodxf.sx . . . . . 6 𝑥𝑆
5048, 49nfeq 2290 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5150a1i 9 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
523, 44, 4, 51sbciedf 2947 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5313, 52mpbid 146 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
542, 53eqtrd 2173 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1332  wnf 1437  wcel 1481  wnfc 2269  Vcvv 2689  [wsbc 2912  (class class class)co 5781  cmpo 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786
This theorem is referenced by:  ovmpodx  5904  mpoxopoveq  6144
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