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Theorem fliftfuns 7318
Description: The function 𝐹 is the unique function defined by 𝐹𝐴 = 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 2892 . . . . 5 𝑦𝐴, 𝐵
3 nfcsb1v 3909 . . . . . 6 𝑥𝑦 / 𝑥𝐴
4 nfcsb1v 3909 . . . . . 6 𝑥𝑦 / 𝑥𝐵
53, 4nfop 4885 . . . . 5 𝑥𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵
6 csbeq1a 3898 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
7 csbeq1a 3898 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
86, 7opeq12d 4877 . . . . 5 (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
92, 5, 8cbvmpt 5254 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
109rneqi 5933 . . 3 ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
111, 10eqtri 2753 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
12 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
1312ralrimiva 3136 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑅)
143nfel1 2909 . . . 4 𝑥𝑦 / 𝑥𝐴𝑅
156eleq1d 2810 . . . 4 (𝑥 = 𝑦 → (𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1614, 15rspc 3589 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1713, 16mpan9 505 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑅)
18 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
1918ralrimiva 3136 . . 3 (𝜑 → ∀𝑥𝑋 𝐵𝑆)
204nfel1 2909 . . . 4 𝑥𝑦 / 𝑥𝐵𝑆
217eleq1d 2810 . . . 4 (𝑥 = 𝑦 → (𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2220, 21rspc 3589 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2319, 22mpan9 505 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐵𝑆)
24 csbeq1 3887 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
25 csbeq1 3887 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
2611, 17, 23, 24, 25fliftfun 7316 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3051  csb 3884  cop 4630  cmpt 5226  ran crn 5673  Fun wfun 6537
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by: (None)
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