Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpodv2 Structured version   Visualization version   GIF version

Theorem ovmpodv2 7303
 Description: Alternate deduction version of ovmpo 7305, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpodv2.1 (𝜑𝐴𝐶)
ovmpodv2.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpodv2.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpodv2.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
Assertion
Ref Expression
ovmpodv2 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpodv2
StepHypRef Expression
1 eqidd 2759 . . 3 (𝜑 → (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpodv2.1 . . . 4 (𝜑𝐴𝐶)
3 ovmpodv2.2 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
4 ovmpodv2.3 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
5 ovmpodv2.4 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
65eqeq2d 2769 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
76biimpd 232 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
8 nfmpo1 7228 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
9 nfcv 2919 . . . . . 6 𝑥𝐴
10 nfcv 2919 . . . . . 6 𝑥𝐵
119, 8, 10nfov 7180 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1211nfeq1 2934 . . . 4 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
13 nfmpo2 7229 . . . 4 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
14 nfcv 2919 . . . . . 6 𝑦𝐴
15 nfcv 2919 . . . . . 6 𝑦𝐵
1614, 13, 15nfov 7180 . . . . 5 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1716nfeq1 2934 . . . 4 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpodf 7301 . . 3 (𝜑 → ((𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
191, 18mpd 15 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
20 oveq 7156 . . 3 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
2120eqeq1d 2760 . 2 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
2219, 21syl5ibrcom 250 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  (class class class)co 7150   ∈ cmpo 7152 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155 This theorem is referenced by:  coaval  17394  xpcco  17499  marrepval  21262  marrepeval  21263  marepveval  21268  submaval  21281  submaeval  21282  minmar1val  21348  minmar1eval  21349  nbgrval  27225
 Copyright terms: Public domain W3C validator