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Mirrors > Home > MPE Home > Th. List > funcnv | Structured version Visualization version GIF version |
Description: The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 6621 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
funcnv | ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3475 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3475 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brelrn 5944 | . . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴) |
4 | 3 | pm4.71ri 560 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) |
5 | 4 | mobii 2538 | . . . 4 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ ∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) |
6 | moanimv 2611 | . . . 4 ⊢ (∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) | |
7 | 5, 6 | bitri 275 | . . 3 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) |
8 | 7 | albii 1814 | . 2 ⊢ (∀𝑦∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) |
9 | funcnv2 6621 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | |
10 | df-ral 3059 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) | |
11 | 8, 9, 10 | 3bitr4i 303 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 ∈ wcel 2099 ∃*wmo 2528 ∀wral 3058 class class class wbr 5148 ◡ccnv 5677 ran crn 5679 Fun wfun 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 |
This theorem is referenced by: funcnv3 6623 fncnv 6626 |
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