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Theorem fliftfuns 5665
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 2256 . . . . 5 𝑦𝐴, 𝐵
3 nfcsb1v 3003 . . . . . 6 𝑥𝑦 / 𝑥𝐴
4 nfcsb1v 3003 . . . . . 6 𝑥𝑦 / 𝑥𝐵
53, 4nfop 3689 . . . . 5 𝑥𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵
6 csbeq1a 2981 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
7 csbeq1a 2981 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
86, 7opeq12d 3681 . . . . 5 (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
92, 5, 8cbvmpt 3991 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
109rneqi 4735 . . 3 ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
111, 10eqtri 2136 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
12 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
1312ralrimiva 2480 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑅)
143nfel1 2267 . . . 4 𝑥𝑦 / 𝑥𝐴𝑅
156eleq1d 2184 . . . 4 (𝑥 = 𝑦 → (𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1614, 15rspc 2755 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1713, 16mpan9 277 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑅)
18 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
1918ralrimiva 2480 . . 3 (𝜑 → ∀𝑥𝑋 𝐵𝑆)
204nfel1 2267 . . . 4 𝑥𝑦 / 𝑥𝐵𝑆
217eleq1d 2184 . . . 4 (𝑥 = 𝑦 → (𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2220, 21rspc 2755 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2319, 22mpan9 277 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐵𝑆)
24 csbeq1 2976 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
25 csbeq1 2976 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
2611, 17, 23, 24, 25fliftfun 5663 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  wral 2391  csb 2973  cop 3498  cmpt 3957  ran crn 4508  Fun wfun 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099
This theorem is referenced by: (None)
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