Theorem fliftrel 5686

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 4566	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
5	2, 3, 4	syl2anc 408	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
6		eqid 2137	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
7	5, 6	fmptd 5567	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑅 × 𝑆))
8		frn 5276	. . 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	eqsstrid 3138	1 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ⊆ wss 3066  ⟨cop 3525   ↦ cmpt 3984   × cxp 4532  ran crn 4535  ⟶wf 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126

This theorem is referenced by:  fliftcnv  5689  fliftfun  5690  fliftf  5693  qliftrel  6501