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Theorem funcnv3 6556

Description: A condition showing a class is single-rooted. (See funcnv 6555). (Contributed by NM, 26-May-2006.)

**Assertion**
Ref	Expression
funcnv3	⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)

Distinct variable group: 𝑥,𝑦,𝐴

**Proof of Theorem funcnv3**
Step	Hyp	Ref	Expression
1		dfrn2 5835	. . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2	1	eqabri 2871	. . . . 5 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
3	2	biimpi 216	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
4	3	biantrurd 532	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → (∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦)))
5	4	ralbiia 3073	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦))
6		funcnv 6555	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7		df-reu 3346	. . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
8		vex 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3442	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	breldm 5855	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴)
11	10	pm4.71ri 560	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
12	11	eubii 2578	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
13		df-eu 2562	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
14	7, 12, 13	3bitr2i 299	. . 3 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
15	14	ralbii 3075	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
16	5, 6, 15	3bitr4i 303	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2109 ∃*wmo 2531 ∃!weu 2561 ∀wral 3044 ∃!wreu 3343 class class class wbr 5095 ^◡ccnv 5622 dom cdm 5623 ran crn 5624 Fun wfun 6480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-br 5096 df-opab 5158 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-fun 6488

This theorem is referenced by: cantnf2 43298

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