Theorem funimaexg 5465

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 5394	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
3		resres 5075	. . . . . . 7 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
4		incom 3421	. . . . . . . 8 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵)
5	4	reseq2i 5060	. . . . . . 7 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
6	3, 5	eqtr4i 2262	. . . . . 6 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7		resdm 5102	. . . . . . 7 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
8	7	reseq1d 5062	. . . . . 6 ⊢ (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵))
9	6, 8	eqtr3id 2285	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))
10	9	rneqd 5011	. . . 4 ⊢ (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4787	. . . 4 ⊢ (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12		df-ima 4787	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
13	10, 11, 12	3eqtr4g 2296	. . 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 4269	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16		inss2 3452	. . . 4 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17		funimaexglem 5464	. . . 4 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
18	16, 17	mp3an3 1367	. . 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 2316	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  dom cdm 4774  ran crn 4775   ↾ cres 4776   “ cima 4777  Rel wrel 4779  Fun wfun 5371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-fun 5379

This theorem is used by:  funimaex  5466  resfunexg  5936  resfunexgALT  6337  fnexALT  6340  suplocexprlem2b  8082  suplocexprlemlub  8092