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Theorem fliftfund 6560
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 1284 . . . 4 (𝜑 → (𝑥𝑋 → (𝑦𝑋 → (𝐴 = 𝐶𝐵 = 𝐷))))
32imp32 449 . . 3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 = 𝐶𝐵 = 𝐷))
43ralrimivva 2970 . 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 6559 . 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 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  cop 4181  cmpt 4727  ran crn 5113  Fun wfun 5880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894
This theorem is referenced by:  cygznlem2a  19910  pi1xfrf  22847  pi1cof  22853
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