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Theorem fliftfund 7060
 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)
fliftfun.4 (𝑥 = 𝑦𝐴 = 𝐶)
fliftfun.5 (𝑥 = 𝑦𝐵 = 𝐷)
fliftfund.6 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝐴 = 𝐶)) → 𝐵 = 𝐷)
Assertion
Ref Expression
fliftfund (𝜑 → Fun 𝐹)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝑦,𝑅   𝑥,𝐷   𝑦,𝐹   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)   𝐹(𝑥)

Proof of Theorem fliftfund
StepHypRef Expression
1 fliftfund.6 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝐴 = 𝐶)) → 𝐵 = 𝐷)
213exp2 1351 . . . 4 (𝜑 → (𝑥𝑋 → (𝑦𝑋 → (𝐴 = 𝐶𝐵 = 𝐷))))
32imp32 422 . . 3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 = 𝐶𝐵 = 𝐷))
43ralrimivva 3120 . 2 (𝜑 → ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))
5 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
6 flift.2 . . 3 ((𝜑𝑥𝑋) → 𝐴𝑅)
7 flift.3 . . 3 ((𝜑𝑥𝑋) → 𝐵𝑆)
8 fliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐶)
9 fliftfun.5 . . 3 (𝑥 = 𝑦𝐵 = 𝐷)
105, 6, 7, 8, 9fliftfun 7059 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
114, 10mpbird 260 1 (𝜑 → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟨cop 4528   ↦ cmpt 5112  ran crn 5525  Fun wfun 6329 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 5169  ax-nul 5176  ax-pr 5298 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 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343 This theorem is referenced by:  cygznlem2a  20335  pi1xfrf  23754  pi1cof  23760
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