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

Theorem xpsfval 16432
Description: The value of the function appearing in xpsval 16437. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
Assertion
Ref Expression
xpsfval ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = ({𝑋} +𝑐 {𝑌}))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem xpsfval
StepHypRef Expression
1 sneq 4380 . . . 4 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2 sneq 4380 . . . 4 (𝑦 = 𝑌 → {𝑦} = {𝑌})
31, 2oveqan12d 6893 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
43cnveqd 5499 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
5 xpsff1o.f . 2 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
6 ovex 6906 . . 3 ({𝑋} +𝑐 {𝑌}) ∈ V
76cnvex 7343 . 2 ({𝑋} +𝑐 {𝑌}) ∈ V
84, 5, 7ovmpt2a 7021 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = ({𝑋} +𝑐 {𝑌}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  {csn 4370  ccnv 5310  (class class class)co 6874  cmpt2 6876   +𝑐 ccda 9274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879
This theorem is referenced by:  xpsff1o  16433  xpsaddlem  16440  xpsvsca  16444  xpsle  16446  xpsdsval  22399
  Copyright terms: Public domain W3C validator