Theorem qliftfund 8799

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)
qliftfund.6	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
qliftfund	⊢ (𝜑 → Fun 𝐹)

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 qliftfund

Step	Hyp	Ref	Expression
1		qliftfund.6	. . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵)
2	1	ex 412	. . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝐴 = 𝐵))
3	2	alrimivv 1923	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))
4		qlift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
5		qlift.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
6		qlift.3	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
7		qlift.4	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
8		qliftfun.4	. . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
9	4, 5, 6, 7, 8	qliftfun 8798	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))
10	3, 9	mpbird 257	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ⟨cop 4629   class class class wbr 5141   ↦ cmpt 5224  ran crn 5670  Fun wfun 6531   Er wer 8702  [cec 8703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-er 8705  df-ec 8707  df-qs 8711

This theorem is referenced by:  orbstafun  19227  frgpupf  19693