Theorem elrn3 36152

Description: Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)

Assertion

Ref	Expression
elrn3	⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Proof of Theorem elrn3

Step	Hyp	Ref	Expression
1		df-rn 5673	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
2	1	eleq2i 2861	. 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵)
3		eldm3 36151	. 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅)
4		cnvxp 6155	. . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V)
5	4	ineq2i 4178	. . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V))
6		cnvin 6142	. . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴}))
7		df-res 5674	. . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V))
8	5, 6, 7	3eqtr4ri 2803	. . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴}))
9	8	eqeq1i 2774	. . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅)
10		relinxp 5802	. . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴}))
11		cnveq0 6197	. . . . 5 ⊢ (Rel (𝐵 ∩ (V × {𝐴})) → ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅))
12	10, 11	ax-mp 5	. . . 4 ⊢ ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅)
13	9, 12	bitr4i 281	. . 3 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅)
14	13	necon3bii 3016	. 2 ⊢ ((◡𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
15	2, 3, 14	3bitri 300	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463   ∩ cin 3912  ∅c0 4294  {csn 4594   × cxp 5660  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   ↾ cres 5664  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674

This theorem is referenced by: (None)