Theorem relmptopab 6264

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

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relmptopab.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ 𝜑})
2	1	funmpt2 5396	. . . . . . 7 ⊢ Fun 𝐹
3		funrel 5374	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Rel 𝐹
5		relelfvdm 5707	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑟 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹)
6	4, 5	mpan 424	. . . . 5 ⊢ (𝑟 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)
7		relopab 4886	. . . . . . 7 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
8		df-rel 4761	. . . . . . 7 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
9	7, 8	mpbi 145	. . . . . 6 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
10	9	rgenw 2599	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
11	1	fvmptssdm 5767	. . . . 5 ⊢ ((𝐵 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)) → (𝐹‘𝐵) ⊆ (V × V))
12	6, 10, 11	sylancl 413	. . . 4 ⊢ (𝑟 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ⊆ (V × V))
13		ssel 3236	. . . 4 ⊢ ((𝐹‘𝐵) ⊆ (V × V) → (𝑟 ∈ (𝐹‘𝐵) → 𝑟 ∈ (V × V)))
14	12, 13	mpcom 36	. . 3 ⊢ (𝑟 ∈ (𝐹‘𝐵) → 𝑟 ∈ (V × V))
15	14	ssriv 3246	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
16		df-rel 4761	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
17	15, 16	mpbir 146	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ⊆ wss 3214  {copab 4175   ↦ cmpt 4176   × cxp 4752  dom cdm 4754  Rel wrel 4759  Fun wfun 5351  ‘cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fv 5365

This theorem is referenced by:  reldvdsr  14336  lmrel  15182  relwlk  16468  reltrls  16503  releupth  16565