Theorem relmpoopab 8077

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 5799	. . . . 5 ⊢ Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑}
2		df-rel 5659	. . . . 5 ⊢ (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V))
3	1, 2	mpbi 233	. . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
4	3	rgen2w 3084	. . 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 8076	. . 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 5659	. 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V))
9	7, 8	mpbir 234	1 ⊢ Rel (𝐶𝐹𝐷)

Syntax hints:   = wceq 1563  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  {copab 5167   × cxp 5650  Rel wrel 5657  (class class class)co 7400   ∈ cmpo 7402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975

This theorem is referenced by:  brovmpoex  8207  relfunc  17909  releqg  19232  relup  49812