Theorem fliftrel 5375

Description: 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)
flift.2	⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)
flift.3	⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)

Assertion

Ref	Expression
fliftrel	⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆))

Distinct variable groups:   x,𝑅   φ,x   x,𝑋   x,𝑆

Allowed substitution hints:   A(x)   B(x)   𝐹(x)

Proof of Theorem fliftrel

Step	Hyp	Ref	Expression
1		flift.1	. 2 ⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)
2		flift.2	. . . . 5 ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)
3		flift.3	. . . . 5 ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)
4		opelxpi 4319	. . . . 5 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → ⟨A, B⟩ ∈ (𝑅 × 𝑆))
5	2, 3, 4	syl2anc 391	. . . 4 ⊢ ((φ ∧ x ∈ 𝑋) → ⟨A, B⟩ ∈ (𝑅 × 𝑆))
6		eqid 2037	. . . 4 ⊢ (x ∈ 𝑋 ↦ ⟨A, B⟩) = (x ∈ 𝑋 ↦ ⟨A, B⟩)
7	5, 6	fmptd 5265	. . 3 ⊢ (φ → (x ∈ 𝑋 ↦ ⟨A, B⟩):𝑋⟶(𝑅 × 𝑆))
8		frn 4995	. . 3 ⊢ ((x ∈ 𝑋 ↦ ⟨A, B⟩):𝑋⟶(𝑅 × 𝑆) → ran (x ∈ 𝑋 ↦ ⟨A, B⟩) ⊆ (𝑅 × 𝑆))
9	7, 8	syl 14	. 2 ⊢ (φ → ran (x ∈ 𝑋 ↦ ⟨A, B⟩) ⊆ (𝑅 × 𝑆))
10	1, 9	syl5eqss 2983	1 ⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  ⟨cop 3370   ↦ cmpt 3809   × cxp 4286  ran crn 4289  ⟶wf 4841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853

This theorem is referenced by:  fliftcnv  5378  fliftfun  5379  fliftf  5382  qliftrel  6121