Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovmpordx Structured version   Visualization version   GIF version

Theorem ovmpordx 46505
Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7509. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
ovmpordx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpordx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpordx.3 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpordx.4 (𝜑𝐴𝐿)
ovmpordx.5 (𝜑𝐵𝐷)
ovmpordx.6 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpordx (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑦,𝐴   𝑥,𝐵   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpordx
StepHypRef Expression
1 ovmpordx.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpordx.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 ovmpordx.3 . 2 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
4 ovmpordx.4 . 2 (𝜑𝐴𝐿)
5 ovmpordx.5 . 2 (𝜑𝐵𝐷)
6 ovmpordx.6 . 2 (𝜑𝑆𝑋)
7 nfv 1918 . 2 𝑥𝜑
8 nfv 1918 . 2 𝑦𝜑
9 nfcv 2904 . 2 𝑦𝐴
10 nfcv 2904 . 2 𝑥𝐵
11 nfcv 2904 . 2 𝑥𝑆
12 nfcv 2904 . 2 𝑦𝑆
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpordxf 46504 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  (class class class)co 7361  cmpo 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366
This theorem is referenced by:  ovmpox2  46506
  Copyright terms: Public domain W3C validator