Theorem elrn3 31848

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 5197	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
2	1	eleq2i 2763	. 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ◡𝐵)
3		eldm3 31847	. 2 ⊢ (𝐴 ∈ dom ◡𝐵 ↔ (◡𝐵 ↾ {𝐴}) ≠ ∅)
4		cnvxp 5629	. . . . . . 7 ⊢ ◡(V × {𝐴}) = ({𝐴} × V)
5	4	ineq2i 3887	. . . . . 6 ⊢ (◡𝐵 ∩ ◡(V × {𝐴})) = (◡𝐵 ∩ ({𝐴} × V))
6		cnvin 5618	. . . . . 6 ⊢ ◡(𝐵 ∩ (V × {𝐴})) = (◡𝐵 ∩ ◡(V × {𝐴}))
7		df-res 5198	. . . . . 6 ⊢ (◡𝐵 ↾ {𝐴}) = (◡𝐵 ∩ ({𝐴} × V))
8	5, 6, 7	3eqtr4ri 2725	. . . . 5 ⊢ (◡𝐵 ↾ {𝐴}) = ◡(𝐵 ∩ (V × {𝐴}))
9	8	eqeq1i 2697	. . . 4 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅)
10		inss2 3910	. . . . . 6 ⊢ (𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴})
11		relxp 5203	. . . . . 6 ⊢ Rel (V × {𝐴})
12		relss 5283	. . . . . 6 ⊢ ((𝐵 ∩ (V × {𝐴})) ⊆ (V × {𝐴}) → (Rel (V × {𝐴}) → Rel (𝐵 ∩ (V × {𝐴}))))
13	10, 11, 12	mp2 9	. . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴}))
14		cnveq0 5669	. . . . 5 ⊢ (Rel (𝐵 ∩ (V × {𝐴})) → ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅))
15	13, 14	ax-mp 5	. . . 4 ⊢ ((𝐵 ∩ (V × {𝐴})) = ∅ ↔ ◡(𝐵 ∩ (V × {𝐴})) = ∅)
16	9, 15	bitr4i 267	. . 3 ⊢ ((◡𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅)
17	16	necon3bii 2916	. 2 ⊢ ((◡𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
18	2, 3, 17	3bitri 286	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Syntax hints:   ↔ wb 196   = wceq 1564   ∈ wcel 2071   ≠ wne 2864  Vcvv 3272   ∩ cin 3647   ⊆ wss 3648  ∅c0 3991  {csn 4253   × cxp 5184  ^◡ccnv 5185  dom cdm 5186  ran crn 5187   ↾ cres 5188  Rel wrel 5191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pr 4979

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-sn 4254  df-pr 4256  df-op 4260  df-br 4729  df-opab 4789  df-xp 5192  df-rel 5193  df-cnv 5194  df-dm 5196  df-rn 5197  df-res 5198

This theorem is referenced by: (None)