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Theorem funimaexg 5207
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 108 . . 3 ((Fun 𝐴𝐵𝐶) → Fun 𝐴)
2 funrel 5140 . . 3 (Fun 𝐴 → Rel 𝐴)
3 resres 4831 . . . . . . 7 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴𝐵))
4 incom 3268 . . . . . . . 8 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
54reseq2i 4816 . . . . . . 7 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴𝐵))
63, 5eqtr4i 2163 . . . . . 6 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7 resdm 4858 . . . . . . 7 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
87reseq1d 4818 . . . . . 6 (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴𝐵))
96, 8syl5eqr 2186 . . . . 5 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
109rneqd 4768 . . . 4 (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴𝐵))
11 df-ima 4552 . . . 4 (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12 df-ima 4552 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
1310, 11, 123eqtr4g 2197 . . 3 (Rel 𝐴 → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
141, 2, 133syl 17 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
15 inex1g 4064 . . 3 (𝐵𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16 inss2 3297 . . . 4 (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17 funimaexglem 5206 . . . 4 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1816, 17mp3an3 1304 . . 3 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1915, 18sylan2 284 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
2014, 19eqeltrrd 2217 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2686  cin 3070  wss 3071  dom cdm 4539  ran crn 4540  cres 4541  cima 4542  Rel wrel 4544  Fun wfun 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125
This theorem is referenced by:  funimaex  5208  resfunexg  5641  resfunexgALT  6008  fnexALT  6011  suplocexprlem2b  7529  suplocexprlemlub  7539
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