Theorem relmptopab 7683

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 7028	. . 3 ⊢ (∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V))
3		relopab 5837	. . . . 5 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4		df-rel 5696	. . . . 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 3065	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
8		df-rel 5696	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
9	7, 8	mpbir 231	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  {copab 5210   ↦ cmpt 5231   × cxp 5687  Rel wrel 5694  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571

This theorem is referenced by:  reldvdsr  20377  lmrel  23254  phtpcrel  25039  ulmrel  26436  ercgrg  28540  relwlk  29659  reltrls  29727  relpths  29753  releupth  30228  acycgr0v  35133  prclisacycgr  35136