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Theorem funimaexg 4926
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 A B 𝐶) → (AB) V)

Proof of Theorem funimaexg
StepHypRef Expression
1 simpl 102 . . 3 ((Fun A B 𝐶) → Fun A)
2 funrel 4862 . . 3 (Fun A → Rel A)
3 resres 4567 . . . . . . 7 ((A ↾ dom A) ↾ B) = (A ↾ (dom AB))
4 incom 3123 . . . . . . . 8 (B ∩ dom A) = (dom AB)
54reseq2i 4552 . . . . . . 7 (A ↾ (B ∩ dom A)) = (A ↾ (dom AB))
63, 5eqtr4i 2060 . . . . . 6 ((A ↾ dom A) ↾ B) = (A ↾ (B ∩ dom A))
7 resdm 4592 . . . . . . 7 (Rel A → (A ↾ dom A) = A)
87reseq1d 4554 . . . . . 6 (Rel A → ((A ↾ dom A) ↾ B) = (AB))
96, 8syl5eqr 2083 . . . . 5 (Rel A → (A ↾ (B ∩ dom A)) = (AB))
109rneqd 4506 . . . 4 (Rel A → ran (A ↾ (B ∩ dom A)) = ran (AB))
11 df-ima 4301 . . . 4 (A “ (B ∩ dom A)) = ran (A ↾ (B ∩ dom A))
12 df-ima 4301 . . . 4 (AB) = ran (AB)
1310, 11, 123eqtr4g 2094 . . 3 (Rel A → (A “ (B ∩ dom A)) = (AB))
141, 2, 133syl 17 . 2 ((Fun A B 𝐶) → (A “ (B ∩ dom A)) = (AB))
15 inex1g 3884 . . 3 (B 𝐶 → (B ∩ dom A) V)
16 inss2 3152 . . . 4 (B ∩ dom A) ⊆ dom A
17 funimaexglem 4925 . . . 4 ((Fun A (B ∩ dom A) V (B ∩ dom A) ⊆ dom A) → (A “ (B ∩ dom A)) V)
1816, 17mp3an3 1220 . . 3 ((Fun A (B ∩ dom A) V) → (A “ (B ∩ dom A)) V)
1915, 18sylan2 270 . 2 ((Fun A B 𝐶) → (A “ (B ∩ dom A)) V)
2014, 19eqeltrrd 2112 1 ((Fun A B 𝐶) → (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  wss 2911  dom cdm 4288  ran crn 4289  cres 4290  cima 4291  Rel wrel 4293  Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by:  funimaex  4927  resfunexg  5325  resfunexgALT  5679  fnexALT  5682
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