funcnv3 - Intuitionistic Logic Explorer

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Theorem funcnv3 5417

Description: A condition showing a class is single-rooted. (See funcnv 5416). (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 4942	. . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2	1	abeq2i 2343	. . . . 5 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
3	2	biimpi 120	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
4	3	biantrurd 305	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → (∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦)))
5	4	ralbiia 2556	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦))
6		funcnv 5416	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7		df-reu 2527	. . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
8		vex 2815	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 2815	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	breldm 4959	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴)
11	10	pm4.71ri 392	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
12	11	eubii 2089	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
13		eu5 2128	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
14	7, 12, 13	3bitr2i 208	. . 3 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
15	14	ralbii 2548	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
16	5, 6, 15	3bitr4i 212	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 ∃wex 1541 ∃!weu 2080 ∃*wmo 2081 ∈ wcel 2203 ∀wral 2520 ∃!wreu 2522 class class class wbr 4108 ^◡ccnv 4747 dom cdm 4748 ran crn 4749 Fun wfun 5345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-reu 2527 df-v 2814 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-br 4109 df-opab 4171 df-id 4413 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-fun 5353

This theorem is referenced by: (None)

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