Theorem relmptopab 7675

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 7020	. . 3 ⊢ (∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V))
3		relopab 5828	. . . . 5 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4		df-rel 5687	. . . . 5 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
5	3, 4	mpbi 229	. . . 4 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
6	5	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
7	2, 6	mprg 3063	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
8		df-rel 5687	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
9	7, 8	mpbir 230	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3471   ⊆ wss 3947  {copab 5212   ↦ cmpt 5233   × cxp 5678  Rel wrel 5685  ‘cfv 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fv 6559

This theorem is referenced by:  reldvdsr  20304  lmrel  23152  phtpcrel  24937  ulmrel  26332  ercgrg  28339  relwlk  29458  reltrls  29526  relpths  29552  releupth  30027  acycgr0v  34763  prclisacycgr  34766