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Theorem funimaexg 4983
 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 102 . . 3 ((Fun 𝐴𝐵𝐶) → Fun 𝐴)
2 funrel 4919 . . 3 (Fun 𝐴 → Rel 𝐴)
3 resres 4624 . . . . . . 7 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴𝐵))
4 incom 3129 . . . . . . . 8 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
54reseq2i 4609 . . . . . . 7 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴𝐵))
63, 5eqtr4i 2063 . . . . . 6 ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7 resdm 4649 . . . . . . 7 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
87reseq1d 4611 . . . . . 6 (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴𝐵))
96, 8syl5eqr 2086 . . . . 5 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
109rneqd 4563 . . . 4 (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴𝐵))
11 df-ima 4358 . . . 4 (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12 df-ima 4358 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
1310, 11, 123eqtr4g 2097 . . 3 (Rel 𝐴 → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
141, 2, 133syl 17 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
15 inex1g 3893 . . 3 (𝐵𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16 inss2 3158 . . . 4 (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17 funimaexglem 4982 . . . 4 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1816, 17mp3an3 1221 . . 3 ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
1915, 18sylan2 270 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
2014, 19eqeltrrd 2115 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∩ cin 2916   ⊆ wss 2917  dom cdm 4345  ran crn 4346   ↾ cres 4347   “ cima 4348  Rel wrel 4350  Fun wfun 4896 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904 This theorem is referenced by:  funimaex  4984  resfunexg  5382  resfunexgALT  5737  fnexALT  5740
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