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Theorem funimaexg 5011
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 106 . . 3 ((Fun 𝐴𝐵𝐶) → Fun 𝐴)
2 funrel 4947 . . 3 (Fun 𝐴 → Rel 𝐴)
3 resres 4652 . . . . . . 7 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴𝐵))
4 incom 3157 . . . . . . . 8 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
54reseq2i 4637 . . . . . . 7 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴𝐵))
63, 5eqtr4i 2079 . . . . . 6 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7 resdm 4677 . . . . . . 7 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
87reseq1d 4639 . . . . . 6 (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴𝐵))
96, 8syl5eqr 2102 . . . . 5 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
109rneqd 4591 . . . 4 (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴𝐵))
11 df-ima 4386 . . . 4 (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12 df-ima 4386 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
1310, 11, 123eqtr4g 2113 . . 3 (Rel 𝐴 → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
141, 2, 133syl 17 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
15 inex1g 3921 . . 3 (𝐵𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16 inss2 3186 . . . 4 (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17 funimaexglem 5010 . . . 4 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1816, 17mp3an3 1232 . . 3 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1915, 18sylan2 274 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
2014, 19eqeltrrd 2131 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cin 2944  wss 2945  dom cdm 4373  ran crn 4374  cres 4375  cima 4376  Rel wrel 4378  Fun wfun 4924
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-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972
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-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932
This theorem is referenced by:  funimaex  5012  resfunexg  5410  resfunexgALT  5765  fnexALT  5768
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