Theorem funimaexg 5302

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) → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem funimaexg

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun 𝐴)
2		funrel 5235	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
3		resres 4921	. . . . . . 7 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
4		incom 3329	. . . . . . . 8 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵)
5	4	reseq2i 4906	. . . . . . 7 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
6	3, 5	eqtr4i 2201	. . . . . 6 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7		resdm 4948	. . . . . . 7 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
8	7	reseq1d 4908	. . . . . 6 ⊢ (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵))
9	6, 8	eqtr3id 2224	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4858	. . . 4 ⊢ (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4641	. . . 4 ⊢ (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12		df-ima 4641	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
13	10, 11, 12	3eqtr4g 2235	. . 3 ⊢ (Rel 𝐴 → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴 “ 𝐵))
14	1, 2, 13	3syl 17	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴 “ 𝐵))
15		inex1g 4141	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16		inss2 3358	. . . 4 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17		funimaexglem 5301	. . . 4 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
18	16, 17	mp3an3 1326	. . 3 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
19	15, 18	sylan2 286	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
20	14, 19	eqeltrrd 2255	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∩ cin 3130   ⊆ wss 3131  dom cdm 4628  ran crn 4629   ↾ cres 4630   “ cima 4631  Rel wrel 4633  Fun wfun 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220

This theorem is referenced by:  funimaex  5303  resfunexg  5739  resfunexgALT  6111  fnexALT  6114  suplocexprlem2b  7715  suplocexprlemlub  7725