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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > alrmomorn | Structured version Visualization version GIF version |
Description: Equivalence of an "at most one" and an "at most one" restricted to the range inside a universal quantification. (Contributed by Peter Mazsa, 3-Sep-2021.) |
Ref | Expression |
---|---|
alrmomorn | ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3352 | . . 3 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) | |
2 | cnvresrn 36740 | . . . . . 6 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | |
3 | 2 | breqi 5110 | . . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ 𝑦◡𝑅𝑥) |
4 | brres 5941 | . . . . . . 7 ⊢ (𝑥 ∈ V → (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥))) | |
5 | 4 | elv 3450 | . . . . . 6 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥)) |
6 | brcnvg 5832 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) | |
7 | 6 | el2v 3452 | . . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
8 | 7 | anbi2i 624 | . . . . . 6 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) |
9 | 5, 8 | bitri 275 | . . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) |
10 | 3, 9, 7 | 3bitr3i 301 | . . . 4 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦) |
11 | 10 | mobii 2548 | . . 3 ⊢ (∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦) |
12 | 1, 11 | bitri 275 | . 2 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦) |
13 | 12 | albii 1822 | 1 ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 ∃*wmo 2538 ∃*wrmo 3351 Vcvv 3444 class class class wbr 5104 ◡ccnv 5630 ran crn 5632 ↾ cres 5633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rmo 3352 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 |
This theorem is referenced by: ineccnvmo2 36752 |
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