Theorem qliftfund 8785

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 416	. . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝐴 = 𝐵))
3	2	alrimivv 1948	. 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 8784	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))
10	3, 9	mpbird 259	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181  ran crn 5648  Fun wfun 6515   Er wer 8675  [cec 8676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-er 8678  df-ec 8680  df-qs 8684

This theorem is referenced by:  orbstafun  19351  frgpupf  19813