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

Theorem xpsfval 16148
 Description: The value of the function appearing in xpsval 16153. (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 4158 . . . 4 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2 sneq 4158 . . . 4 (𝑦 = 𝑌 → {𝑦} = {𝑌})
31, 2oveqan12d 6623 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
43cnveqd 5258 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
5 xpsff1o.f . 2 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
6 ovex 6632 . . 3 ({𝑋} +𝑐 {𝑌}) ∈ V
76cnvex 7060 . 2 ({𝑋} +𝑐 {𝑌}) ∈ V
84, 5, 7ovmpt2a 6744 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = ({𝑋} +𝑐 {𝑌}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {csn 4148  ◡ccnv 5073  (class class class)co 6604   ↦ cmpt2 6606   +𝑐 ccda 8933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609 This theorem is referenced by:  xpsff1o  16149  xpsaddlem  16156  xpsvsca  16160  xpsle  16162  xpsdsval  22096
 Copyright terms: Public domain W3C validator