elrn3 - Mathbox for Scott Fenton

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Theorem elrn3 35351

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 5684	. . 3 ⊢ ran 𝐵 = dom ^◡𝐵
2	1	eleq2i 2821	. 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ 𝐴 ∈ dom ^◡𝐵)
3		eldm3 35350	. 2 ⊢ (𝐴 ∈ dom ^◡𝐵 ↔ (^◡𝐵 ↾ {𝐴}) ≠ ∅)
4		cnvxp 6156	. . . . . . 7 ⊢ ^◡(V × {𝐴}) = ({𝐴} × V)
5	4	ineq2i 4206	. . . . . 6 ⊢ (^◡𝐵 ∩ ^◡(V × {𝐴})) = (^◡𝐵 ∩ ({𝐴} × V))
6		cnvin 6144	. . . . . 6 ⊢ ^◡(𝐵 ∩ (V × {𝐴})) = (^◡𝐵 ∩ ^◡(V × {𝐴}))
7		df-res 5685	. . . . . 6 ⊢ (^◡𝐵 ↾ {𝐴}) = (^◡𝐵 ∩ ({𝐴} × V))
8	5, 6, 7	3eqtr4ri 2767	. . . . 5 ⊢ (^◡𝐵 ↾ {𝐴}) = ^◡(𝐵 ∩ (V × {𝐴}))
9	8	eqeq1i 2733	. . . 4 ⊢ ((^◡𝐵 ↾ {𝐴}) = ∅ ↔ ^◡(𝐵 ∩ (V × {𝐴})) = ∅)
10		relinxp 5811	. . . . 5 ⊢ Rel (𝐵 ∩ (V × {𝐴}))
11		cnveq0 6196	. . . . 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 278	. . 3 ⊢ ((^◡𝐵 ↾ {𝐴}) = ∅ ↔ (𝐵 ∩ (V × {𝐴})) = ∅)
14	13	necon3bii 2989	. 2 ⊢ ((^◡𝐵 ↾ {𝐴}) ≠ ∅ ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
15	2, 3, 14	3bitri 297	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ∩ cin 3944 ∅c0 4319 {csn 4625 × cxp 5671 ^◡ccnv 5672 dom cdm 5673 ran crn 5674 ↾ cres 5675 Rel wrel 5678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-opab 5206 df-xp 5679 df-rel 5680 df-cnv 5681 df-dm 5683 df-rn 5684 df-res 5685

This theorem is referenced by: (None)

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