Theorem fliftel 7345

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 5167	. . 3 ⊢ (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2		flift.1	. . . 4 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
3	2	eleq2i 2836	. . 3 ⊢ (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
4		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
5		opex 5484	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6	4, 5	elrnmpti 5985	. . 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 5499	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
11	8, 9, 10	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
12	11	rexbidva 3183	. 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 1537   ∈ wcel 2108  ∃wrex 3076  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  fliftcnv  7347  fliftfun  7348  fliftf  7351  fliftval  7352  qliftel  8858