Theorem funimaexg 2715

Description: Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.

Assertion

Ref	Expression
funimaexg	⊢ (B ∈ C → (Fun A → (A “ B) ∈ V))

Proof of Theorem funimaexg

Step	Hyp	Ref	Expression
1		imaeq2 2603	. . . 4 ⊢ (w = B → (A “ w) = (A “ B))
2	1	eleq1d 1155	. . 3 ⊢ (w = B → ((A “ w) ∈ V ↔ (A “ B) ∈ V))
3	2	imbi2d 464	. 2 ⊢ (w = B → ((Fun A → (A “ w) ∈ V) ↔ (Fun A → (A “ B) ∈ V)))
4		dffun5 2677	. . . 4 ⊢ (Fun A ↔ (Rel A ∧ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)))
5	4	pm3.27bd 263	. . 3 ⊢ (Fun A → ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z))
6		ax-17 925	. . . . 5 ⊢ (⟨x, y⟩ ∈ A → ∀z⟨x, y⟩ ∈ A)
7	6	zfrep2 1475	. . . 4 ⊢ (∀x∃z∀y(⟨x, y⟩ ∈ A → y = z) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)))
8		isset 1351	. . . . 5 ⊢ ((A “ w) ∈ V ↔ ∃z z = (A “ w))
9		dfima3 2605	. . . . . . . 8 ⊢ (A “ w) = {y∣∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)}
10	9	cleq2i 1111	. . . . . . 7 ⊢ (z = (A “ w) ↔ z = {y∣∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)})
11		cleqab 1174	. . . . . . 7 ⊢ (z = {y∣∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)} ↔ ∀y(y ∈ z ↔ ∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)))
12	10, 11	bitr 151	. . . . . 6 ⊢ (z = (A “ w) ↔ ∀y(y ∈ z ↔ ∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)))
13	12	biex 733	. . . . 5 ⊢ (∃z z = (A “ w) ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)))
14	8, 13	bitr 151	. . . 4 ⊢ ((A “ w) ∈ V ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ ⟨x, y⟩ ∈ A)))
15	7, 14	sylibr 175	. . 3 ⊢ (∀x∃z∀y(⟨x, y⟩ ∈ A → y = z) → (A “ w) ∈ V)
16	5, 15	syl 12	. 2 ⊢ (Fun A → (A “ w) ∈ V)
17	3, 16	vtoclg 1383	1 ⊢ (B ∈ C → (Fun A → (A “ B) ∈ V))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   “ cima 2413  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  funimaex 2716  resfunexg 2717  fnex 2740  carduniima 3695

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432

metamath.org