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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦}))

Assertion

Ref	Expression
xpsfval	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (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

Step	Hyp	Ref	Expression
1		sneq 4326	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
2		sneq 4326	. . . 4 ⊢ (𝑦 = 𝑌 → {𝑦} = {𝑌})
3	1, 2	oveqan12d 6812	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
4	3	cnveqd 5436	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ◡({𝑥} +𝑐 {𝑦}) = ◡({𝑋} +𝑐 {𝑌}))
5		xpsff1o.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦}))
6		ovex 6823	. . 3 ⊢ ({𝑋} +𝑐 {𝑌}) ∈ V
7	6	cnvex 7260	. 2 ⊢ ◡({𝑋} +𝑐 {𝑌}) ∈ V
8	4, 5, 7	ovmpt2a 6938	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = ◡({𝑋} +𝑐 {𝑌}))

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