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

Theorem ovmpodv2 7586
Description: Alternate deduction version of ovmpo 7588, 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 2729 . . 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 2739 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
76biimpd 228 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
8 nfmpo1 7507 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
9 nfcv 2899 . . . . . 6 𝑥𝐴
10 nfcv 2899 . . . . . 6 𝑥𝐵
119, 8, 10nfov 7456 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1211nfeq1 2915 . . . 4 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
13 nfmpo2 7508 . . . 4 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
14 nfcv 2899 . . . . . 6 𝑦𝐴
15 nfcv 2899 . . . . . 6 𝑦𝐵
1614, 13, 15nfov 7456 . . . . 5 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1716nfeq1 2915 . . . 4 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpodf 7584 . . 3 (𝜑 → ((𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
191, 18mpd 15 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
20 oveq 7432 . . 3 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
2120eqeq1d 2730 . 2 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
2219, 21syl5ibrcom 246 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  (class class class)co 7426  cmpo 7428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431
This theorem is referenced by:  coaval  18066  xpcco  18183  marrepval  22492  marrepeval  22493  marepveval  22498  submaval  22511  submaeval  22512  minmar1val  22578  minmar1eval  22579  nbgrval  29177  clnbgrval  47209
  Copyright terms: Public domain W3C validator