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Theorem fliftf 5382
Description: The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
flift.2 ((φ x 𝑋) → A 𝑅)
flift.3 ((φ x 𝑋) → B 𝑆)
Assertion
Ref Expression
fliftf (φ → (Fun 𝐹𝐹:ran (x 𝑋A)⟶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 103 . . . . 5 ((φ Fun 𝐹) → Fun 𝐹)
2 flift.1 . . . . . . . . . . 11 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
3 flift.2 . . . . . . . . . . 11 ((φ x 𝑋) → A 𝑅)
4 flift.3 . . . . . . . . . . 11 ((φ x 𝑋) → B 𝑆)
52, 3, 4fliftel 5376 . . . . . . . . . 10 (φ → (y𝐹zx 𝑋 (y = A z = B)))
65exbidv 1703 . . . . . . . . 9 (φ → (z y𝐹zzx 𝑋 (y = A z = B)))
76adantr 261 . . . . . . . 8 ((φ Fun 𝐹) → (z y𝐹zzx 𝑋 (y = A z = B)))
8 rexcom4 2571 . . . . . . . . 9 (x 𝑋 z(y = A z = B) ↔ zx 𝑋 (y = A z = B))
9 elisset 2562 . . . . . . . . . . . . . 14 (B 𝑆z z = B)
104, 9syl 14 . . . . . . . . . . . . 13 ((φ x 𝑋) → z z = B)
1110biantrud 288 . . . . . . . . . . . 12 ((φ x 𝑋) → (y = A ↔ (y = A z z = B)))
12 19.42v 1783 . . . . . . . . . . . 12 (z(y = A z = B) ↔ (y = A z z = B))
1311, 12syl6rbbr 188 . . . . . . . . . . 11 ((φ x 𝑋) → (z(y = A z = B) ↔ y = A))
1413rexbidva 2317 . . . . . . . . . 10 (φ → (x 𝑋 z(y = A z = B) ↔ x 𝑋 y = A))
1514adantr 261 . . . . . . . . 9 ((φ Fun 𝐹) → (x 𝑋 z(y = A z = B) ↔ x 𝑋 y = A))
168, 15syl5bbr 183 . . . . . . . 8 ((φ Fun 𝐹) → (zx 𝑋 (y = A z = B) ↔ x 𝑋 y = A))
177, 16bitrd 177 . . . . . . 7 ((φ Fun 𝐹) → (z y𝐹zx 𝑋 y = A))
1817abbidv 2152 . . . . . 6 ((φ Fun 𝐹) → {yz y𝐹z} = {yx 𝑋 y = A})
19 df-dm 4298 . . . . . 6 dom 𝐹 = {yz y𝐹z}
20 eqid 2037 . . . . . . 7 (x 𝑋A) = (x 𝑋A)
2120rnmpt 4525 . . . . . 6 ran (x 𝑋A) = {yx 𝑋 y = A}
2218, 19, 213eqtr4g 2094 . . . . 5 ((φ Fun 𝐹) → dom 𝐹 = ran (x 𝑋A))
23 df-fn 4848 . . . . 5 (𝐹 Fn ran (x 𝑋A) ↔ (Fun 𝐹 dom 𝐹 = ran (x 𝑋A)))
241, 22, 23sylanbrc 394 . . . 4 ((φ Fun 𝐹) → 𝐹 Fn ran (x 𝑋A))
252, 3, 4fliftrel 5375 . . . . . . 7 (φ𝐹 ⊆ (𝑅 × 𝑆))
2625adantr 261 . . . . . 6 ((φ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27 rnss 4507 . . . . . 6 (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
2826, 27syl 14 . . . . 5 ((φ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29 rnxpss 4697 . . . . 5 ran (𝑅 × 𝑆) ⊆ 𝑆
3028, 29syl6ss 2951 . . . 4 ((φ Fun 𝐹) → ran 𝐹𝑆)
31 df-f 4849 . . . 4 (𝐹:ran (x 𝑋A)⟶𝑆 ↔ (𝐹 Fn ran (x 𝑋A) ran 𝐹𝑆))
3224, 30, 31sylanbrc 394 . . 3 ((φ Fun 𝐹) → 𝐹:ran (x 𝑋A)⟶𝑆)
3332ex 108 . 2 (φ → (Fun 𝐹𝐹:ran (x 𝑋A)⟶𝑆))
34 ffun 4991 . 2 (𝐹:ran (x 𝑋A)⟶𝑆 → Fun 𝐹)
3533, 34impbid1 130 1 (φ → (Fun 𝐹𝐹:ran (x 𝑋A)⟶𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301  wss 2911  cop 3370   class class class wbr 3755  cmpt 3809   × cxp 4286  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by:  qliftf  6127
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