Theorem funimaexg 3571

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

Assertion

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

Proof of Theorem funimaexg

Step	Hyp	Ref	Expression
1		imaeq2 3398	. . . . 5 ⊢ (w = B → (A “ w) = (A “ B))
2	1	eleq1d 1538	. . . 4 ⊢ (w = B → ((A “ w) ∈ V ↔ (A “ B) ∈ V))
3	2	imbi2d 611	. . 3 ⊢ (w = B → ((Fun A → (A “ w) ∈ V) ↔ (Fun A → (A “ B) ∈ V)))
4		dffun5 3525	. . . . 5 ⊢ (Fun A ↔ (Rel A ⋀ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)))
5	4	pm3.27bi 326	. . . 4 ⊢ (Fun A → ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z))
6		ax-17 970	. . . . . 6 ⊢ (⟨x, y⟩ ∈ A → ∀z⟨x, y⟩ ∈ A)
7	6	axrep4 2693	. . . . 5 ⊢ (∀x∃z∀y(⟨x, y⟩ ∈ A → y = z) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)))
8		isset 1811	. . . . . 6 ⊢ ((A “ w) ∈ V ↔ ∃z z = (A “ w))
9		dfima3 3402	. . . . . . . . 9 ⊢ (A “ w) = {y∣∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)}
10	9	eqeq2i 1483	. . . . . . . 8 ⊢ (z = (A “ w) ↔ z = {y∣∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)})
11		abeq2 1566	. . . . . . . 8 ⊢ (z = {y∣∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)} ↔ ∀y(y ∈ z ↔ ∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)))
12	10, 11	bitr 173	. . . . . . 7 ⊢ (z = (A “ w) ↔ ∀y(y ∈ z ↔ ∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)))
13	12	exbii 1050	. . . . . 6 ⊢ (∃z z = (A “ w) ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)))
14	8, 13	bitr 173	. . . . 5 ⊢ ((A “ w) ∈ V ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ⋀ ⟨x, y⟩ ∈ A)))
15	7, 14	sylibr 200	. . . 4 ⊢ (∀x∃z∀y(⟨x, y⟩ ∈ A → y = z) → (A “ w) ∈ V)
16	5, 15	syl 10	. . 3 ⊢ (Fun A → (A “ w) ∈ V)
17	3, 16	vtoclg 1844	. 2 ⊢ (B ∈ C → (Fun A → (A “ B) ∈ V))
18	17	impcom 351	1 ⊢ ((Fun A ⋀ B ∈ C) → (A “ B) ∈ V)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  ∃wex 979  {cab 1462  Vcvv 1808  ⟨cop 2408   “ cima 3169  Rel wrel 3171  Fun wfun 3172

This theorem is referenced by:  funimaex 3572  resfunexg 3575  fnex 3603  carduniima 4873

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188

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