Theorem qliftfun 8809

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 8805	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6		eceq1 8743	. . 3 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		qliftfun.4	. . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
8	1, 5, 2, 6, 7	fliftfun 7321	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
9	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
11	9, 10	ercl 8715	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝑋)
12	9, 10	ercl2 8717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝑋)
13	11, 12	jca 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
14	13	ex 418	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
15	14	pm4.71rd 572	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦)))
16	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 Er 𝑋)
17		simprl 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
18	16, 17	erth 8758	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
19	18	pm5.32da 590	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
20	15, 19	bitrd 282	. . . . . 6 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
21	20	imbi1d 344	. . . . 5 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22		impexp 456	. . . . 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 290	. . . 4 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
24	23	2albidv 1956	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
25		r2al 3204	. . 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 292	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
27	8, 26	bitr4d 285	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ⟨cop 4600   class class class wbr 5114   ↦ cmpt 5197  ran crn 5667  Fun wfun 6537   Er wer 8700  [cec 8701   / cqs 8702

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-er 8703  df-ec 8705  df-qs 8709

This theorem is used by:  qliftfund  8810  qliftfuns  8811