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Theorem qliftf 6090
Description: The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
qlift.2 ((φ x 𝑋) → A 𝑌)
qlift.3 (φ𝑅 Er 𝑋)
qlift.4 (φ𝑋 V)
Assertion
Ref Expression
qliftf (φ → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
Distinct variable groups:   φ,x   x,𝑅   x,𝑋   x,𝑌
Allowed substitution hints:   A(x)   𝐹(x)

Proof of Theorem qliftf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
2 qlift.2 . . . 4 ((φ x 𝑋) → A 𝑌)
3 qlift.3 . . . 4 (φ𝑅 Er 𝑋)
4 qlift.4 . . . 4 (φ𝑋 V)
51, 2, 3, 4qliftlem 6083 . . 3 ((φ x 𝑋) → [x]𝑅 (𝑋 / 𝑅))
61, 5, 2fliftf 5352 . 2 (φ → (Fun 𝐹𝐹:ran (x 𝑋 ↦ [x]𝑅)⟶𝑌))
7 df-qs 6011 . . . . 5 (𝑋 / 𝑅) = {yx 𝑋 y = [x]𝑅}
8 eqid 2013 . . . . . 6 (x 𝑋 ↦ [x]𝑅) = (x 𝑋 ↦ [x]𝑅)
98rnmpt 4497 . . . . 5 ran (x 𝑋 ↦ [x]𝑅) = {yx 𝑋 y = [x]𝑅}
107, 9eqtr4i 2036 . . . 4 (𝑋 / 𝑅) = ran (x 𝑋 ↦ [x]𝑅)
1110a1i 9 . . 3 (φ → (𝑋 / 𝑅) = ran (x 𝑋 ↦ [x]𝑅))
1211feq2d 4949 . 2 (φ → (𝐹:(𝑋 / 𝑅)⟶𝑌𝐹:ran (x 𝑋 ↦ [x]𝑅)⟶𝑌))
136, 12bitr4d 180 1 (φ → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1223   wcel 1366  {cab 1999  wrex 2276  Vcvv 2526  cop 3342  cmpt 3781  ran crn 4261  Fun wfun 4811  wf 4813   Er wer 6002  [cec 6003   / cqs 6004
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-un 4108
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-rab 2284  df-v 2528  df-sbc 2733  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-fv 4825  df-er 6005  df-ec 6007  df-qs 6011
This theorem is referenced by: (None)
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