Theorem fvdiagfn 8864

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 4599	. . 3 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	xpeq2d 5668	. 2 ⊢ (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
4		simpr 484	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
5		snex 5391	. . . 4 ⊢ {𝑋} ∈ V
6		xpexg 7726	. . . 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 6991	1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {csn 4589   ↦ cmpt 5188   × cxp 5636  ‘cfv 6511

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519

This theorem is referenced by:  pwsdiagmhm  18758  pwsdiaglmhm  20964