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Theorem fliftfuns 7314
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 2902 . . . . 5 𝑦𝐴, 𝐵
3 nfcsb1v 3918 . . . . . 6 𝑥𝑦 / 𝑥𝐴
4 nfcsb1v 3918 . . . . . 6 𝑥𝑦 / 𝑥𝐵
53, 4nfop 4889 . . . . 5 𝑥𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵
6 csbeq1a 3907 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
7 csbeq1a 3907 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
86, 7opeq12d 4881 . . . . 5 (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
92, 5, 8cbvmpt 5259 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
109rneqi 5936 . . 3 ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
111, 10eqtri 2759 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
12 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
1312ralrimiva 3145 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑅)
143nfel1 2918 . . . 4 𝑥𝑦 / 𝑥𝐴𝑅
156eleq1d 2817 . . . 4 (𝑥 = 𝑦 → (𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1614, 15rspc 3600 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1713, 16mpan9 506 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑅)
18 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
1918ralrimiva 3145 . . 3 (𝜑 → ∀𝑥𝑋 𝐵𝑆)
204nfel1 2918 . . . 4 𝑥𝑦 / 𝑥𝐵𝑆
217eleq1d 2817 . . . 4 (𝑥 = 𝑦 → (𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2220, 21rspc 3600 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2319, 22mpan9 506 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐵𝑆)
24 csbeq1 3896 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
25 csbeq1 3896 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
2611, 17, 23, 24, 25fliftfun 7312 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 395   = wceq 1540  wcel 2105  wral 3060  csb 3893  cop 4634  cmpt 5231  ran crn 5677  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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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