funcnv3 - Intuitionistic Logic Explorer

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Theorem funcnv3 5441

Description: A condition showing a class is single-rooted. (See funcnv 5440). (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 4966	. . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2	1	abeq2i 2349	. . . . 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 2564	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦))
6		funcnv 5440	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7		df-reu 2535	. . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
8		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 2824	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	breldm 4983	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴)
11	10	pm4.71ri 396	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
12	11	eubii 2095	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
13		eu5 2134	. . . 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 2556	. 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 1545 ∃!weu 2086 ∃*wmo 2087 ∈ wcel 2209 ∀wral 2528 ∃!wreu 2530 class class class wbr 4128 ^◡ccnv 4771 dom cdm 4772 ran crn 4773 Fun wfun 5369

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 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-reu 2535 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-br 4129 df-opab 4191 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-fun 5377

This theorem is referenced by: (None)

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