Theorem relmptopab 6781

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 6201	. . 3 ⊢ (∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V))
3		relopab 5169	. . . . 5 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4		df-rel 5045	. . . . 5 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
5	3, 4	mpbi 219	. . . 4 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
6	5	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
7	2, 6	mprg 2910	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
8		df-rel 5045	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
9	7, 8	mpbir 220	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {copab 4642   ↦ cmpt 4643   × cxp 5036  Rel wrel 5043  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812

This theorem is referenced by:  reldvdsr  18467  lmrel  20844  phtpcrel  22600  ulmrel  23936  ercgrg  25212  rel1wlk  40830  releupth  41366