Theorem relmptopab 7639

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
relmptopab.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Assertion

Ref	Expression
relmptopab	⊢ Rel (𝐹‘𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem relmptopab

Step	Hyp	Ref	Expression
1		relmptopab.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ 𝜑})
2	1	fvmptss 6980	. . 3 ⊢ (∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V))
3		relopab 5787	. . . . 5 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4		df-rel 5645	. . . . 5 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
5	3, 4	mpbi 230	. . . 4 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
6	5	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
7	2, 6	mprg 3050	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
8		df-rel 5645	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
9	7, 8	mpbir 231	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  {copab 5169   ↦ cmpt 5188   × cxp 5636  Rel wrel 5643  ‘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-pr 5387

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-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  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-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519

This theorem is referenced by:  reldvdsr  20269  lmrel  23117  phtpcrel  24892  ulmrel  26287  ercgrg  28444  relwlk  29554  reltrls  29622  relpths  29648  releupth  30128  acycgr0v  35135  prclisacycgr  35138