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Theorem funcnv3 6118

Description: A condition showing a class is single-rooted. (See funcnv 6117). (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 5464	. . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2	1	abeq2i 2871	. . . . 5 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
3	2	biimpi 206	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
4	3	biantrurd 530	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → (∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦)))
5	4	ralbiia 3115	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃𝑥 𝑥𝐴𝑦))
6		funcnv 6117	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7		df-reu 3055	. . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
8		vex 3341	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3341	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	breldm 5482	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴)
11	10	pm4.71ri 668	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
12	11	eubii 2627	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦))
13		eu5 2631	. . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
14	7, 12, 13	3bitr2i 288	. . 3 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
15	14	ralbii 3116	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
16	5, 6, 15	3bitr4i 292	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 383 ∃wex 1851 ∈ wcel 2137 ∃!weu 2605 ∃*wmo 2606 ∀wral 3048 ∃!wreu 3050 class class class wbr 4802 ^◡ccnv 5263 dom cdm 5264 ran crn 5265 Fun wfun 6041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pr 5053

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ral 3053 df-reu 3055 df-rab 3057 df-v 3340 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-sn 4320 df-pr 4322 df-op 4326 df-br 4803 df-opab 4863 df-id 5172 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-fun 6049

This theorem is referenced by: (None)

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