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Mirrors > Home > MPE Home > Th. List > funcnv3 | Structured version Visualization version GIF version |
Description: A condition showing a class is single-rooted. (See funcnv 6627). (Contributed by NM, 26-May-2006.) |
Ref | Expression |
---|---|
funcnv3 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 5895 | . . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
2 | 1 | eqabri 2873 | . . . . 5 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦) |
3 | 2 | biimpi 215 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦) |
4 | 3 | biantrurd 531 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))) |
5 | 4 | ralbiia 3088 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
6 | funcnv 6627 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) | |
7 | df-reu 3375 | . . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) | |
8 | vex 3477 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
9 | vex 3477 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | breldm 5915 | . . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴) |
11 | 10 | pm4.71ri 559 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) |
12 | 11 | eubii 2574 | . . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) |
13 | df-eu 2558 | . . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) | |
14 | 7, 12, 13 | 3bitr2i 298 | . . 3 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
15 | 14 | ralbii 3090 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
16 | 5, 6, 15 | 3bitr4i 302 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∃wex 1773 ∈ wcel 2098 ∃*wmo 2527 ∃!weu 2557 ∀wral 3058 ∃!wreu 3372 class class class wbr 5152 ◡ccnv 5681 dom cdm 5682 ran crn 5683 Fun wfun 6547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-fun 6555 |
This theorem is referenced by: cantnf2 42785 |
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