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Theorem fliftfun 5463
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 1437 . . 3 𝑥𝜑
2 flift.1 . . . . 5 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
3 nfmpt1 3877 . . . . . 6 𝑥(𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
43nfrn 4606 . . . . 5 𝑥ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
52, 4nfcxfr 2191 . . . 4 𝑥𝐹
65nffun 4951 . . 3 𝑥Fun 𝐹
7 fveq2 5205 . . . . . . 7 (𝐴 = 𝐶 → (𝐹𝐴) = (𝐹𝐶))
8 simplr 490 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → Fun 𝐹)
9 flift.2 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐴𝑅)
10 flift.3 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐵𝑆)
112, 9, 10fliftel1 5461 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
1211ad2ant2r 486 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝐹𝐵)
13 funbrfv 5239 . . . . . . . . 9 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
148, 12, 13sylc 60 . . . . . . . 8 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝐴) = 𝐵)
15 simprr 492 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
16 eqidd 2057 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 = 𝐶)
17 eqidd 2057 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 = 𝐷)
18 fliftfun.4 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 = 𝐶)
1918eqeq2d 2067 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐶 = 𝐴𝐶 = 𝐶))
20 fliftfun.5 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐵 = 𝐷)
2120eqeq2d 2067 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐷 = 𝐵𝐷 = 𝐷))
2219, 21anbi12d 450 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
2322rspcev 2673 . . . . . . . . . . 11 ((𝑦𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
2415, 16, 17, 23syl12anc 1144 . . . . . . . . . 10 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
252, 9, 10fliftel 5460 . . . . . . . . . . 11 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
2625ad2antrr 465 . . . . . . . . . 10 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
2724, 26mpbird 160 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶𝐹𝐷)
28 funbrfv 5239 . . . . . . . . 9 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
298, 27, 28sylc 60 . . . . . . . 8 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝐶) = 𝐷)
3014, 29eqeq12d 2070 . . . . . . 7 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐵 = 𝐷))
317, 30syl5ib 147 . . . . . 6 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 = 𝐶𝐵 = 𝐷))
3231anassrs 386 . . . . 5 ((((𝜑 ∧ Fun 𝐹) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝐴 = 𝐶𝐵 = 𝐷))
3332ralrimiva 2409 . . . 4 (((𝜑 ∧ Fun 𝐹) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))
3433exp31 350 . . 3 (𝜑 → (Fun 𝐹 → (𝑥𝑋 → ∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))))
351, 6, 34ralrimd 2414 . 2 (𝜑 → (Fun 𝐹 → ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
362, 9, 10fliftel 5460 . . . . . . . . 9 (𝜑 → (𝑧𝐹𝑢 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵)))
372, 9, 10fliftel 5460 . . . . . . . . . 10 (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑣 = 𝐵)))
3818eqeq2d 2067 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑧 = 𝐴𝑧 = 𝐶))
3920eqeq2d 2067 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑣 = 𝐵𝑣 = 𝐷))
4038, 39anbi12d 450 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑣 = 𝐵) ↔ (𝑧 = 𝐶𝑣 = 𝐷)))
4140cbvrexv 2551 . . . . . . . . . 10 (∃𝑥𝑋 (𝑧 = 𝐴𝑣 = 𝐵) ↔ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))
4237, 41syl6bb 189 . . . . . . . . 9 (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)))
4336, 42anbi12d 450 . . . . . . . 8 (𝜑 → ((𝑧𝐹𝑢𝑧𝐹𝑣) ↔ (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))))
4443biimpd 136 . . . . . . 7 (𝜑 → ((𝑧𝐹𝑢𝑧𝐹𝑣) → (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))))
45 reeanv 2496 . . . . . . . 8 (∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) ↔ (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)))
46 r19.29 2467 . . . . . . . . . 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 2467 . . . . . . . . . . . 12 ((∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → ∃𝑦𝑋 ((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))))
48 eqtr2 2074 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝐴𝑧 = 𝐶) → 𝐴 = 𝐶)
4948ad2ant2r 486 . . . . . . . . . . . . . . . 16 (((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝐴 = 𝐶)
5049imim1i 58 . . . . . . . . . . . . . . 15 ((𝐴 = 𝐶𝐵 = 𝐷) → (((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝐵 = 𝐷))
5150imp 119 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝐵 = 𝐷)
52 simprlr 498 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝐵)
53 simprrr 500 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑣 = 𝐷)
5451, 52, 533eqtr4d 2098 . . . . . . . . . . . . 13 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5554rexlimivw 2446 . . . . . . . . . . . 12 (∃𝑦𝑋 ((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5647, 55syl 14 . . . . . . . . . . 11 ((∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5756rexlimivw 2446 . . . . . . . . . 10 (∃𝑥𝑋 (∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5846, 57syl 14 . . . . . . . . 9 ((∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5958ex 112 . . . . . . . 8 (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → (∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝑢 = 𝑣))
6045, 59syl5bir 146 . . . . . . 7 (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ((∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)) → 𝑢 = 𝑣))
6144, 60syl9 70 . . . . . 6 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6261alrimdv 1772 . . . . 5 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6362alrimdv 1772 . . . 4 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6463alrimdv 1772 . . 3 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
652, 9, 10fliftrel 5459 . . . . 5 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
66 relxp 4474 . . . . 5 Rel (𝑅 × 𝑆)
67 relss 4454 . . . . 5 (𝐹 ⊆ (𝑅 × 𝑆) → (Rel (𝑅 × 𝑆) → Rel 𝐹))
6865, 66, 67mpisyl 1351 . . . 4 (𝜑 → Rel 𝐹)
69 dffun2 4939 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7069baib 839 . . . 4 (Rel 𝐹 → (Fun 𝐹 ↔ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7168, 70syl 14 . . 3 (𝜑 → (Fun 𝐹 ↔ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7264, 71sylibrd 162 . 2 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → Fun 𝐹))
7335, 72impbid 124 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2944  cop 3405   class class class wbr 3791  cmpt 3845   × cxp 4370  ran crn 4373  Rel wrel 4377  Fun wfun 4923  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937
This theorem is referenced by:  fliftfund  5464  fliftfuns  5465  qliftfun  6218
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