Theorem fliftel 7243

Description: Elementhood in the relation 𝐹. (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
fliftel	⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))

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

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

Proof of Theorem fliftel

Step	Hyp	Ref	Expression
1		df-br 5092	. . 3 ⊢ (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2		flift.1	. . . 4 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
3	2	eleq2i 2823	. . 3 ⊢ (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
4		eqid 2731	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
5		opex 5404	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	elrnmpti 5902	. . 3 ⊢ (⟨𝐶, 𝐷⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ↔ ∃𝑥 ∈ 𝑋 ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
7	1, 3, 6	3bitri 297	. 2 ⊢ (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
8		flift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
9		flift.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
10		opthg2 5419	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
11	8, 9, 10	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
12	11	rexbidva 3154	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
13	7, 12	bitrid 283	1 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ⟨cop 4582   class class class wbr 5091   ↦ cmpt 5172  ran crn 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-mpt 5173  df-cnv 5624  df-dm 5626  df-rn 5627

This theorem is referenced by:  fliftcnv  7245  fliftfun  7246  fliftf  7249  fliftval  7250  qliftel  8724