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Theorem fliftfuns 7067
 Description: The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2 ((𝜑𝑥𝑋) → 𝐴𝑅)
flift.3 ((𝜑𝑥𝑋) → 𝐵𝑆)
Assertion
Ref Expression
fliftfuns (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑥,𝑧,𝑦,𝑅   𝑦,𝐹,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftfuns
StepHypRef Expression
1 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
2 nfcv 2919 . . . . 5 𝑦𝐴, 𝐵
3 nfcsb1v 3831 . . . . . 6 𝑥𝑦 / 𝑥𝐴
4 nfcsb1v 3831 . . . . . 6 𝑥𝑦 / 𝑥𝐵
53, 4nfop 4782 . . . . 5 𝑥𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵
6 csbeq1a 3821 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
7 csbeq1a 3821 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
86, 7opeq12d 4774 . . . . 5 (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
92, 5, 8cbvmpt 5137 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
109rneqi 5783 . . 3 ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
111, 10eqtri 2781 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
12 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
1312ralrimiva 3113 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑅)
143nfel1 2935 . . . 4 𝑥𝑦 / 𝑥𝐴𝑅
156eleq1d 2836 . . . 4 (𝑥 = 𝑦 → (𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1614, 15rspc 3531 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1713, 16mpan9 510 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑅)
18 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
1918ralrimiva 3113 . . 3 (𝜑 → ∀𝑥𝑋 𝐵𝑆)
204nfel1 2935 . . . 4 𝑥𝑦 / 𝑥𝐵𝑆
217eleq1d 2836 . . . 4 (𝑥 = 𝑦 → (𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2220, 21rspc 3531 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2319, 22mpan9 510 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐵𝑆)
24 csbeq1 3810 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
25 csbeq1 3810 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
2611, 17, 23, 24, 25fliftfun 7065 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⦋csb 3807  ⟨cop 4531   ↦ cmpt 5116  ran crn 5529  Fun wfun 6334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348 This theorem is referenced by: (None)
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