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Theorem fliftf 5466
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 107 . . . . 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 5460 . . . . . . . . . 10 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
65exbidv 1722 . . . . . . . . 9 (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
76adantr 265 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
8 rexcom4 2594 . . . . . . . . 9 (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵))
9 elisset 2585 . . . . . . . . . . . . . 14 (𝐵𝑆 → ∃𝑧 𝑧 = 𝐵)
104, 9syl 14 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∃𝑧 𝑧 = 𝐵)
1110biantrud 292 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
12 19.42v 1802 . . . . . . . . . . . 12 (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
1311, 12syl6rbbr 192 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
1413rexbidva 2340 . . . . . . . . . 10 (𝜑 → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1514adantr 265 . . . . . . . . 9 ((𝜑 ∧ Fun 𝐹) → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
168, 15syl5bbr 187 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
177, 16bitrd 181 . . . . . . 7 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1817abbidv 2171 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴})
19 df-dm 4382 . . . . . 6 dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20 eqid 2056 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2120rnmpt 4609 . . . . . 6 ran (𝑥𝑋𝐴) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴}
2218, 19, 213eqtr4g 2113 . . . . 5 ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥𝑋𝐴))
23 df-fn 4932 . . . . 5 (𝐹 Fn ran (𝑥𝑋𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥𝑋𝐴)))
241, 22, 23sylanbrc 402 . . . 4 ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥𝑋𝐴))
252, 3, 4fliftrel 5459 . . . . . . 7 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
2625adantr 265 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27 rnss 4591 . . . . . 6 (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
2826, 27syl 14 . . . . 5 ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29 rnxpss 4781 . . . . 5 ran (𝑅 × 𝑆) ⊆ 𝑆
3028, 29syl6ss 2984 . . . 4 ((𝜑 ∧ Fun 𝐹) → ran 𝐹𝑆)
31 df-f 4933 . . . 4 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥𝑋𝐴) ∧ ran 𝐹𝑆))
3224, 30, 31sylanbrc 402 . . 3 ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥𝑋𝐴)⟶𝑆)
3332ex 112 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
34 ffun 5075 . 2 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 → Fun 𝐹)
3533, 34impbid1 134 1 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324  wss 2944  cop 3405   class class class wbr 3791  cmpt 3845   × cxp 4370  dom cdm 4372  ran crn 4373  Fun wfun 4923   Fn wfn 4924  wf 4925
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-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:  qliftf  6221
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