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Theorem fliftfun 6522
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)
Assertion
Ref Expression
fliftfun (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
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 fliftfun
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . 3 𝑥𝜑
2 flift.1 . . . . 5 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
3 nfmpt1 4712 . . . . . 6 𝑥(𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
43nfrn 5333 . . . . 5 𝑥ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
52, 4nfcxfr 2759 . . . 4 𝑥𝐹
65nffun 5875 . . 3 𝑥Fun 𝐹
7 fveq2 6153 . . . . . . 7 (𝐴 = 𝐶 → (𝐹𝐴) = (𝐹𝐶))
8 simplr 791 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → Fun 𝐹)
9 flift.2 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐴𝑅)
10 flift.3 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐵𝑆)
112, 9, 10fliftel1 6520 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
1211ad2ant2r 782 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝐹𝐵)
13 funbrfv 6196 . . . . . . . . 9 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
148, 12, 13sylc 65 . . . . . . . 8 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝐴) = 𝐵)
15 simprr 795 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
16 eqidd 2622 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 = 𝐶)
17 eqidd 2622 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 = 𝐷)
18 fliftfun.4 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 = 𝐶)
1918eqeq2d 2631 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐶 = 𝐴𝐶 = 𝐶))
20 fliftfun.5 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐵 = 𝐷)
2120eqeq2d 2631 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐷 = 𝐵𝐷 = 𝐷))
2219, 21anbi12d 746 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
2322rspcev 3298 . . . . . . . . . . 11 ((𝑦𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
2415, 16, 17, 23syl12anc 1321 . . . . . . . . . 10 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
252, 9, 10fliftel 6519 . . . . . . . . . . 11 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
2625ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
2724, 26mpbird 247 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶𝐹𝐷)
28 funbrfv 6196 . . . . . . . . 9 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
298, 27, 28sylc 65 . . . . . . . 8 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝐶) = 𝐷)
3014, 29eqeq12d 2636 . . . . . . 7 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐵 = 𝐷))
317, 30syl5ib 234 . . . . . 6 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 = 𝐶𝐵 = 𝐷))
3231anassrs 679 . . . . 5 ((((𝜑 ∧ Fun 𝐹) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝐴 = 𝐶𝐵 = 𝐷))
3332ralrimiva 2961 . . . 4 (((𝜑 ∧ Fun 𝐹) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))
3433exp31 629 . . 3 (𝜑 → (Fun 𝐹 → (𝑥𝑋 → ∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))))
351, 6, 34ralrimd 2954 . 2 (𝜑 → (Fun 𝐹 → ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
362, 9, 10fliftel 6519 . . . . . . . . 9 (𝜑 → (𝑧𝐹𝑢 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵)))
372, 9, 10fliftel 6519 . . . . . . . . . 10 (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑣 = 𝐵)))
3818eqeq2d 2631 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑧 = 𝐴𝑧 = 𝐶))
3920eqeq2d 2631 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑣 = 𝐵𝑣 = 𝐷))
4038, 39anbi12d 746 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑣 = 𝐵) ↔ (𝑧 = 𝐶𝑣 = 𝐷)))
4140cbvrexv 3163 . . . . . . . . . 10 (∃𝑥𝑋 (𝑧 = 𝐴𝑣 = 𝐵) ↔ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))
4237, 41syl6bb 276 . . . . . . . . 9 (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)))
4336, 42anbi12d 746 . . . . . . . 8 (𝜑 → ((𝑧𝐹𝑢𝑧𝐹𝑣) ↔ (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))))
4443biimpd 219 . . . . . . 7 (𝜑 → ((𝑧𝐹𝑢𝑧𝐹𝑣) → (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))))
45 reeanv 3100 . . . . . . . 8 (∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) ↔ (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)))
46 r19.29 3066 . . . . . . . . . 10 ((∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → ∃𝑥𝑋 (∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))))
47 r19.29 3066 . . . . . . . . . . . 12 ((∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → ∃𝑦𝑋 ((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))))
48 eqtr2 2641 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝐴𝑧 = 𝐶) → 𝐴 = 𝐶)
4948ad2ant2r 782 . . . . . . . . . . . . . . . 16 (((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝐴 = 𝐶)
5049imim1i 63 . . . . . . . . . . . . . . 15 ((𝐴 = 𝐶𝐵 = 𝐷) → (((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝐵 = 𝐷))
5150imp 445 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝐵 = 𝐷)
52 simprlr 802 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝐵)
53 simprrr 804 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑣 = 𝐷)
5451, 52, 533eqtr4d 2665 . . . . . . . . . . . . 13 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5554rexlimivw 3023 . . . . . . . . . . . 12 (∃𝑦𝑋 ((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5647, 55syl 17 . . . . . . . . . . 11 ((∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5756rexlimivw 3023 . . . . . . . . . 10 (∃𝑥𝑋 (∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5846, 57syl 17 . . . . . . . . 9 ((∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5958ex 450 . . . . . . . 8 (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → (∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝑢 = 𝑣))
6045, 59syl5bir 233 . . . . . . 7 (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ((∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)) → 𝑢 = 𝑣))
6144, 60syl9 77 . . . . . 6 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6261alrimdv 1854 . . . . 5 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6362alrimdv 1854 . . . 4 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6463alrimdv 1854 . . 3 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
652, 9, 10fliftrel 6518 . . . . 5 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
66 relxp 5193 . . . . 5 Rel (𝑅 × 𝑆)
67 relss 5172 . . . . 5 (𝐹 ⊆ (𝑅 × 𝑆) → (Rel (𝑅 × 𝑆) → Rel 𝐹))
6865, 66, 67mpisyl 21 . . . 4 (𝜑 → Rel 𝐹)
69 dffun2 5862 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7069baib 943 . . . 4 (Rel 𝐹 → (Fun 𝐹 ↔ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7168, 70syl 17 . . 3 (𝜑 → (Fun 𝐹 ↔ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7264, 71sylibrd 249 . 2 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → Fun 𝐹))
7335, 72impbid 202 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3559  cop 4159   class class class wbr 4618  cmpt 4678   × cxp 5077  ran crn 5080  Rel wrel 5084  Fun wfun 5846  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860
This theorem is referenced by:  fliftfund  6523  fliftfuns  6524  qliftfun  7784
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