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Theorem fliftf 5668
Description: The domain and range of the function 𝐹. (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)
Assertion
Ref Expression
fliftf (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftf
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5 ((𝜑 ∧ Fun 𝐹) → Fun 𝐹)
2 flift.1 . . . . . . . . . . 11 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
3 flift.2 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐴𝑅)
4 flift.3 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐵𝑆)
52, 3, 4fliftel 5662 . . . . . . . . . 10 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
65exbidv 1781 . . . . . . . . 9 (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
76adantr 274 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
8 rexcom4 2683 . . . . . . . . 9 (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵))
9 elisset 2674 . . . . . . . . . . . . . 14 (𝐵𝑆 → ∃𝑧 𝑧 = 𝐵)
104, 9syl 14 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∃𝑧 𝑧 = 𝐵)
1110biantrud 302 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
12 19.42v 1862 . . . . . . . . . . . 12 (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
1311, 12syl6rbbr 198 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
1413rexbidva 2411 . . . . . . . . . 10 (𝜑 → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1514adantr 274 . . . . . . . . 9 ((𝜑 ∧ Fun 𝐹) → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
168, 15syl5bbr 193 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
177, 16bitrd 187 . . . . . . 7 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1817abbidv 2235 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴})
19 df-dm 4519 . . . . . 6 dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20 eqid 2117 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2120rnmpt 4757 . . . . . 6 ran (𝑥𝑋𝐴) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴}
2218, 19, 213eqtr4g 2175 . . . . 5 ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥𝑋𝐴))
23 df-fn 5096 . . . . 5 (𝐹 Fn ran (𝑥𝑋𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥𝑋𝐴)))
241, 22, 23sylanbrc 413 . . . 4 ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥𝑋𝐴))
252, 3, 4fliftrel 5661 . . . . . . 7 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
2625adantr 274 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27 rnss 4739 . . . . . 6 (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
2826, 27syl 14 . . . . 5 ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29 rnxpss 4940 . . . . 5 ran (𝑅 × 𝑆) ⊆ 𝑆
3028, 29sstrdi 3079 . . . 4 ((𝜑 ∧ Fun 𝐹) → ran 𝐹𝑆)
31 df-f 5097 . . . 4 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥𝑋𝐴) ∧ ran 𝐹𝑆))
3224, 30, 31sylanbrc 413 . . 3 ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥𝑋𝐴)⟶𝑆)
3332ex 114 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
34 ffun 5245 . 2 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 → Fun 𝐹)
3533, 34impbid1 141 1 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wrex 2394  wss 3041  cop 3500   class class class wbr 3899  cmpt 3959   × cxp 4507  dom cdm 4509  ran crn 4510  Fun wfun 5087   Fn wfn 5088  wf 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  qliftf  6482
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