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Theorem funimaexg 4897
 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 4833 . . 3 (Fun A → Rel A)
3 resres 4539 . . . . . . 7 ((A ↾ dom A) ↾ B) = (A ↾ (dom AB))
4 incom 3097 . . . . . . . 8 (B ∩ dom A) = (dom AB)
54reseq2i 4524 . . . . . . 7 (A ↾ (B ∩ dom A)) = (A ↾ (dom AB))
63, 5eqtr4i 2036 . . . . . 6 ((A ↾ dom A) ↾ B) = (A ↾ (B ∩ dom A))
7 resdm 4564 . . . . . . 7 (Rel A → (A ↾ dom A) = A)
87reseq1d 4526 . . . . . 6 (Rel A → ((A ↾ dom A) ↾ B) = (AB))
96, 8syl5eqr 2059 . . . . 5 (Rel A → (A ↾ (B ∩ dom A)) = (AB))
109rneqd 4478 . . . 4 (Rel A → ran (A ↾ (B ∩ dom A)) = ran (AB))
11 df-ima 4273 . . . 4 (A “ (B ∩ dom A)) = ran (A ↾ (B ∩ dom A))
12 df-ima 4273 . . . 4 (AB) = ran (AB)
1310, 11, 123eqtr4g 2070 . . 3 (Rel A → (A “ (B ∩ dom A)) = (AB))
141, 2, 133syl 17 . 2 ((Fun A B 𝐶) → (A “ (B ∩ dom A)) = (AB))
15 inex1g 3856 . . 3 (B 𝐶 → (B ∩ dom A) V)
16 inss2 3126 . . . 4 (B ∩ dom A) ⊆ dom A
17 funimaexglem 4896 . . . 4 ((Fun A (B ∩ dom A) V (B ∩ dom A) ⊆ dom A) → (A “ (B ∩ dom A)) V)
1816, 17mp3an3 1201 . . 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 2088 1 ((Fun A B 𝐶) → (AB) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1223   ∈ wcel 1366  Vcvv 2526   ∩ cin 2884   ⊆ wss 2885  dom cdm 4260  ran crn 4261   ↾ cres 4262   “ cima 4263  Rel wrel 4265  Fun wfun 4811 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-fun 4819 This theorem is referenced by:  funimaex  4898  resfunexg  5295  resfunexgALT  5648  fnexALT  5651
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