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Theorem fliftfund 6709
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)
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 1447 . . . 4 (𝜑 → (𝑥𝑋 → (𝑦𝑋 → (𝐴 = 𝐶𝐵 = 𝐷))))
32imp32 405 . . 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 6708 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
114, 10mpbird 247 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cop 4323  cmpt 4864  ran crn 5251  Fun wfun 6024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038
This theorem is referenced by:  cygznlem2a  20131  pi1xfrf  23072  pi1cof  23078
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