Theorem relmpoopab 8036

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

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

Assertion

Ref	Expression
relmpoopab	⊢ Rel (𝐶𝐹𝐷)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑦,𝐵   𝑥,𝐴,𝑦

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

Proof of Theorem relmpoopab

Step	Hyp	Ref	Expression
1		relopabv 5770	. . . . 5 ⊢ Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑}
2		df-rel 5631	. . . . 5 ⊢ (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V))
3	1, 2	mpbi 230	. . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
4	3	rgen2w 3056	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
5		relmpoopab.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})
6	5	ovmptss 8035	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V) → (𝐶𝐹𝐷) ⊆ (V × V))
7	4, 6	ax-mp 5	. 2 ⊢ (𝐶𝐹𝐷) ⊆ (V × V)
8		df-rel 5631	. 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V))
9	7, 8	mpbir 231	1 ⊢ Rel (𝐶𝐹𝐷)

Syntax hints:   = wceq 1541  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  {copab 5160   × cxp 5622  Rel wrel 5629  (class class class)co 7358   ∈ cmpo 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934

This theorem is referenced by:  brovmpoex  8165  relfunc  17786  releqg  19104  relup  49424