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Theorem xpsfval 16435
Description: The value of the function appearing in xpsval 16440. (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 4326 . . . 4 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2 sneq 4326 . . . 4 (𝑦 = 𝑌 → {𝑦} = {𝑌})
31, 2oveqan12d 6812 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
43cnveqd 5436 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
5 xpsff1o.f . 2 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
6 ovex 6823 . . 3 ({𝑋} +𝑐 {𝑌}) ∈ V
76cnvex 7260 . 2 ({𝑋} +𝑐 {𝑌}) ∈ V
84, 5, 7ovmpt2a 6938 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = ({𝑋} +𝑐 {𝑌}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {csn 4316  ccnv 5248  (class class class)co 6793  cmpt2 6795   +𝑐 ccda 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798
This theorem is referenced by:  xpsff1o  16436  xpsaddlem  16443  xpsvsca  16447  xpsle  16449  xpsdsval  22406
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