Theorem fvdiagfn 8825

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 4589	. . 3 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	xpeq2d 5653	. 2 ⊢ (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
4		simpr 484	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
5		snex 5378	. . . 4 ⊢ {𝑋} ∈ V
6		xpexg 7690	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
7	5, 6	mpan2 691	. . 3 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {𝑋}) ∈ V)
8	7	adantr 480	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐼 × {𝑋}) ∈ V)
9	1, 3, 4, 8	fvmptd3 6957	1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579   ↦ cmpt 5176   × cxp 5621  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494

This theorem is referenced by:  pwsdiagmhm  18723  pwsdiaglmhm  20979