Theorem fvdiagfn 8679

Description: Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypothesis

Ref	Expression
fdiagfn.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))

Assertion

Ref	Expression
fvdiagfn	⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋}))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐼   𝑥,𝑊   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem fvdiagfn

Step	Hyp	Ref	Expression
1		fdiagfn.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))
2		sneq 4571	. . 3 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	xpeq2d 5619	. 2 ⊢ (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
4		simpr 485	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
5		snex 5354	. . . 4 ⊢ {𝑋} ∈ V
6		xpexg 7600	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
7	5, 6	mpan2 688	. . 3 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {𝑋}) ∈ V)
8	7	adantr 481	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐼 × {𝑋}) ∈ V)
9	1, 3, 4, 8	fvmptd3 6898	1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561   ↦ cmpt 5157   × cxp 5587  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by:  pwsdiagmhm  18469  pwsdiaglmhm  20319