Theorem dfrnf 2561

Description: Range definition requiring only that x and y not be 'free' in A (but not necessarily absent from it).

Hypotheses

Ref	Expression
dfrnf.1	⊢ (z ∈ A → ∀x z ∈ A)
dfrnf.2	⊢ (z ∈ A → ∀y z ∈ A)

Assertion

Ref	Expression
dfrnf	⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}

Distinct variable group(s):   x,y,z   z,A

Proof of Theorem dfrnf

Step	Hyp	Ref	Expression
1		dfrn3 2524	. 2 ⊢ ran A = {w∣∃v⟨v, w⟩ ∈ A}
2		ax-17 925	. . . . 5 ⊢ (z ∈ ⟨v, w⟩ → ∀x z ∈ ⟨v, w⟩)
3		dfrnf.1	. . . . 5 ⊢ (z ∈ A → ∀x z ∈ A)
4	2, 3	hbel 1172	. . . 4 ⊢ (⟨v, w⟩ ∈ A → ∀x⟨v, w⟩ ∈ A)
5		ax-17 925	. . . 4 ⊢ (⟨x, w⟩ ∈ A → ∀v⟨x, w⟩ ∈ A)
6		opeq1 1876	. . . . 5 ⊢ (v = x → ⟨v, w⟩ = ⟨x, w⟩)
7	6	eleq1d 1155	. . . 4 ⊢ (v = x → (⟨v, w⟩ ∈ A ↔ ⟨x, w⟩ ∈ A))
8	4, 5, 7	cbvex äSPAN CLASS=r STYLE="color:#FF900D">849	. . 3 ⊢ (∃v⟨v, w⟩ ∈ A ↔ ∃x⟨x, w⟩ ∈ A)
9	8	biabi 1181	. 2 ⊢ {w∣∃v⟨v, w⟩ ∈ A} = {w∣∃x⟨x, w⟩ ∈ A}
10		ax-17 925	. . . . 5 ⊢ (z ∈ ⟨x, w⟩ → ∀y z ∈ ⟨x, w⟩)
11		dfrnf.2	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
12	10, 11	hbel 1172	. . . 4 ⊢ (⟨x, w⟩ ∈ A → ∀y⟨x, w⟩ ∈ A)
13	12	hbex 701	. . 3 ⊢ (∃x⟨x, w⟩ ∈ A → ∀y∃x⟨x, w⟩ ∈ A)
14		ax-17 925	. . 3 ⊢ (∃x⟨x, y⟩ ∈ A → ∀w∃x⟨x, y⟩ ∈ A)
15		opeq2 1877	. . . . 5 ⊢ (w = y → ⟨x, w⟩ = ⟨x, y⟩)
16	15	eleq1d 1155	. . . 4 ⊢ (w = y → (⟨x, w⟩ ∈ A ↔ ⟨x, y⟩ ∈ A))
17	16	biexdv 936	. . 3 ⊢ (w = y → (∃x⟨x, w⟩ ∈ A ↔ ∃x⟨x, y⟩ ∈ A))
18	13, 14, 17	cbvab 1423	. 2 ⊢ {w∣∃x⟨x, w⟩ ∈ A} = {y∣∃x⟨x, y⟩ ∈ A}
19	1, 9, 18	3eqtr 1123	1 ⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  ran crn 2411

This theorem is referenced by:  rnopab 2566

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-clab 1093  df-cleq 1097  df-clel 1099  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-cnv 2426  df-dm 2428  df-rn2429

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