Theorem funimaexg 5132

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 108	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun 𝐴)
2		funrel 5066	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
3		resres 4757	. . . . . . 7 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
4		incom 3207	. . . . . . . 8 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵)
5	4	reseq2i 4742	. . . . . . 7 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
6	3, 5	eqtr4i 2118	. . . . . 6 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7		resdm 4784	. . . . . . 7 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
8	7	reseq1d 4744	. . . . . 6 ⊢ (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵))
9	6, 8	syl5eqr 2141	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4696	. . . 4 ⊢ (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4480	. . . 4 ⊢ (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12		df-ima 4480	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
13	10, 11, 12	3eqtr4g 2152	. . 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 3996	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16		inss2 3236	. . . 4 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17		funimaexglem 5131	. . . 4 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
18	16, 17	mp3an3 1269	. . 3 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
19	15, 18	sylan2 281	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
20	14, 19	eqeltrrd 2172	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∩ cin 3012   ⊆ wss 3013  dom cdm 4467  ran crn 4468   ↾ cres 4469   “ cima 4470  Rel wrel 4472  Fun wfun 5043

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-fun 5051

This theorem is referenced by:  funimaex  5133  resfunexg  5557  resfunexgALT  5919  fnexALT  5922