Theorem funimaexg 5342

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 5275	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
3		resres 4958	. . . . . . 7 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
4		incom 3355	. . . . . . . 8 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵)
5	4	reseq2i 4943	. . . . . . 7 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵))
6	3, 5	eqtr4i 2220	. . . . . 6 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
7		resdm 4985	. . . . . . 7 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
8	7	reseq1d 4945	. . . . . 6 ⊢ (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵))
9	6, 8	eqtr3id 2243	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4895	. . . 4 ⊢ (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4676	. . . 4 ⊢ (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴))
12		df-ima 4676	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
13	10, 11, 12	3eqtr4g 2254	. . 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 4169	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∩ dom 𝐴) ∈ V)
16		inss2 3384	. . . 4 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
17		funimaexglem 5341	. . . 4 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V)
18	16, 17	mp3an3 1337	. . 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 2274	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  dom cdm 4663  ran crn 4664   ↾ cres 4665   “ cima 4666  Rel wrel 4668  Fun wfun 5252

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-fun 5260

This theorem is referenced by:  funimaex  5343  resfunexg  5783  resfunexgALT  6165  fnexALT  6168  suplocexprlem2b  7781  suplocexprlemlub  7791