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Theorem funimaexg 5049
Description: Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
Assertion
Ref Expression
funimaexg ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem funimaexg
StepHypRef Expression
1 simpl 107 . . 3 ((Fun 𝐴𝐵𝐶) → Fun 𝐴)
2 funrel 4984 . . 3 (Fun 𝐴 → Rel 𝐴)
3 resres 4681 . . . . . . 7 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴𝐵))
4 incom 3176 . . . . . . . 8 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
54reseq2i 4666 . . . . . . 7 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴𝐵))
63, 5eqtr4i 2106 . . . . . 6 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7 resdm 4706 . . . . . . 7 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
87reseq1d 4668 . . . . . 6 (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴𝐵))
96, 8syl5eqr 2129 . . . . 5 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
109rneqd 4620 . . . 4 (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴𝐵))
11 df-ima 4412 . . . 4 (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12 df-ima 4412 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
1310, 11, 123eqtr4g 2140 . . 3 (Rel 𝐴 → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
141, 2, 133syl 17 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
15 inex1g 3940 . . 3 (𝐵𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16 inss2 3205 . . . 4 (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17 funimaexglem 5048 . . . 4 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1816, 17mp3an3 1258 . . 3 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1915, 18sylan2 280 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
2014, 19eqeltrrd 2160 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2612  cin 2983  wss 2984  dom cdm 4399  ran crn 4400  cres 4401  cima 4402  Rel wrel 4404  Fun wfun 4961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-fun 4969
This theorem is referenced by:  funimaex  5050  resfunexg  5456  resfunexgALT  5814  fnexALT  5817
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