Theorem xpsfval 16227

Description: The value of the function appearing in xpsval 16232. (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 4187	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
2		sneq 4187	. . . 4 ⊢ (𝑦 = 𝑌 → {𝑦} = {𝑌})
3	1, 2	oveqan12d 6669	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌}))
4	3	cnveqd 5298	. 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 6678	. . 3 ⊢ ({𝑋} +𝑐 {𝑌}) ∈ V
7	6	cnvex 7113	. 2 ⊢ ◡({𝑋} +𝑐 {𝑌}) ∈ V
8	4, 5, 7	ovmpt2a 6791	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = ◡({𝑋} +𝑐 {𝑌}))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177  ^◡ccnv 5113  (class class class)co 6650   ↦ cmpt2 6652   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  xpsff1o  16228  xpsaddlem  16235  xpsvsca  16239  xpsle  16241  xpsdsval  22186