Theorem xpsfval 16050

Description: The value of the function appearing in xpsval 16055. (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 4135	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
2		sneq 4135	. . . 4 ⊢ (𝑦 = 𝑌 → {𝑦} = {𝑌})
3	1, 2	oveqan12d 6568	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
4	3	cnveqd 5220	. 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 6577	. . 3 ⊢ ({𝑋} +𝑐 {𝑌}) ∈ V
7	6	cnvex 7006	. 2 ⊢ ◡({𝑋} +𝑐 {𝑌}) ∈ V
8	4, 5, 7	ovmpt2a 6689	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = ◡({𝑋} +𝑐 {𝑌}))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  ^◡ccnv 5037  (class class class)co 6549   ↦ cmpt2 6551   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  xpsff1o  16051  xpsaddlem  16058  xpsvsca  16062  xpsle  16064  xpsdsval  21996