Theorem relmptopab 6223

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 5365	. . . . . . 7 ⊢ Fun 𝐹
3		funrel 5343	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Rel 𝐹
5		relelfvdm 5671	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑟 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹)
6	4, 5	mpan 424	. . . . 5 ⊢ (𝑟 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)
7		relopab 4856	. . . . . . 7 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
8		df-rel 4732	. . . . . . 7 ⊢ (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
9	7, 8	mpbi 145	. . . . . 6 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
10	9	rgenw 2587	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
11	1	fvmptssdm 5731	. . . . 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 3221	. . . 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 3231	. 2 ⊢ (𝐹‘𝐵) ⊆ (V × V)
16		df-rel 4732	. 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V))
17	15, 16	mpbir 146	1 ⊢ Rel (𝐹‘𝐵)

Syntax hints:   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ⊆ wss 3200  {copab 4149   ↦ cmpt 4150   × cxp 4723  dom cdm 4725  Rel wrel 4730  Fun wfun 5320  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by:  reldvdsr  14104  lmrel  14914  relwlk  16197  reltrls  16232  releupth  16294