Theorem relmptopab 6219

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 5363	. . . . . . 7 ⊢ Fun 𝐹
3		funrel 5341	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Rel 𝐹
5		relelfvdm 5667	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑟 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹)
6	4, 5	mpan 424	. . . . 5 ⊢ (𝑟 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)
7		relopab 4854	. . . . . . 7 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
8		df-rel 4730	. . . . . . 7 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
9	7, 8	mpbi 145	. . . . . 6 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
10	9	rgenw 2585	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
11	1	fvmptssdm 5727	. . . . 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 3219	. . . 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 3229	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
16		df-rel 4730	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
17	15, 16	mpbir 146	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2800   ⊆ wss 3198  {copab 4147   ↦ cmpt 4148   × cxp 4721  dom cdm 4723  Rel wrel 4728  Fun wfun 5318  ‘cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  reldvdsr  14095  lmrel  14905  relwlk  16144  reltrls  16177  releupth  16239