Theorem relmptopab 6234

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 5372	. . . . . . 7 ⊢ Fun 𝐹
3		funrel 5350	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Rel 𝐹
5		relelfvdm 5680	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑟 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹)
6	4, 5	mpan 424	. . . . 5 ⊢ (𝑟 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)
7		relopab 4862	. . . . . . 7 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
8		df-rel 4738	. . . . . . 7 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
9	7, 8	mpbi 145	. . . . . 6 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
10	9	rgenw 2588	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
11	1	fvmptssdm 5740	. . . . 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 3222	. . . 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 3232	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
16		df-rel 4738	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
17	15, 16	mpbir 146	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  {copab 4154   ↦ cmpt 4155   × cxp 4729  dom cdm 4731  Rel wrel 4736  Fun wfun 5327  ‘cfv 5333

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fv 5341

This theorem is referenced by:  reldvdsr  14186  lmrel  15002  relwlk  16288  reltrls  16323  releupth  16385