Theorem funimaexg 6410

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

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq2 5892	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 “ 𝑤) = (𝐴 “ 𝐵))
2	1	eleq1d 2874	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴 “ 𝑤) ∈ V ↔ (𝐴 “ 𝐵) ∈ V))
3	2	imbi2d 344	. . 3 ⊢ (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴 “ 𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)))
4		dffun5 6337	. . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩ ∈ 𝐴
6	5	axrep4 5159	. . . . 5 ⊢ (∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
7		isset 3453	. . . . . 6 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴 “ 𝑤))
8		dfima3 5899	. . . . . . . . 9 ⊢ (𝐴 “ 𝑤) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
9	8	eqeq2i 2811	. . . . . . . 8 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
10		abeq2 2922	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
11	9, 10	bitri 278	. . . . . . 7 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
12	11	exbii 1849	. . . . . 6 ⊢ (∃𝑧 𝑧 = (𝐴 “ 𝑤) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
13	7, 12	bitri 278	. . . . 5 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
14	6, 13	sylibr 237	. . . 4 ⊢ (∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧) → (𝐴 “ 𝑤) ∈ V)
15	4, 14	simplbiim 508	. . 3 ⊢ (Fun 𝐴 → (𝐴 “ 𝑤) ∈ V)
16	3, 15	vtoclg 3515	. 2 ⊢ (𝐵 ∈ 𝐶 → (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V))
17	16	impcom 411	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  Vcvv 3441  ⟨cop 4531   “ cima 5522  Rel wrel 5524  Fun wfun 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326

This theorem is referenced by:  funimaex  6411  resfunexg  6955  resfunexgALT  7631  fnexALT  7634  wdomimag  9035  carduniima  9507  dfac12lem2  9555  ttukeylem3  9922  nnexALT  11627  seqex  13366  fbasrn  22489  elfm3  22555  bdayimaon  33310  nosupno  33316  madeval  33402  fundcmpsurinjlem3  43917  fundcmpsurbijinjpreimafv  43924  fundcmpsurbijinj  43927  fundcmpsurinjALT  43929