Theorem qliftfun 8549

Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)

Hypotheses

Ref	Expression
qlift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
qlift.3	⊢ (𝜑 → 𝑅 Er 𝑋)
qlift.4	⊢ (𝜑 → 𝑋 ∈ 𝑉)
qliftfun.4	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
qliftfun	⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦,𝜑   𝑥,𝑅,𝑦   𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem qliftfun

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
3		qlift.3	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
4		qlift.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
5	1, 2, 3, 4	qliftlem 8545	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6		eceq1 8494	. . 3 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		qliftfun.4	. . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
8	1, 5, 2, 6, 7	fliftfun 7163	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
9	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
11	9, 10	ercl 8467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝑋)
12	9, 10	ercl2 8469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝑋)
13	11, 12	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
14	13	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
15	14	pm4.71rd 562	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦)))
16	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 Er 𝑋)
17		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
18	16, 17	erth 8505	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
19	18	pm5.32da 578	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
20	15, 19	bitrd 278	. . . . . 6 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
21	20	imbi1d 341	. . . . 5 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22		impexp 450	. . . . 5 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
23	21, 22	bitrdi 286	. . . 4 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
24	23	2albidv 1927	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
25		r2al 3124	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
26	24, 25	bitr4di 288	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
27	8, 26	bitr4d 281	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153  ran crn 5581  Fun wfun 6412   Er wer 8453  [cec 8454   / cqs 8455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-er 8456  df-ec 8458  df-qs 8462

This theorem is referenced by:  qliftfund  8550  qliftfuns  8551