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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 3149 | . . 3 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) | |
2 | cnvresrn 35609 | . . . . . 6 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | |
3 | 2 | breqi 5075 | . . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ 𝑦◡𝑅𝑥) |
4 | brres 5863 | . . . . . . 7 ⊢ (𝑥 ∈ V → (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥))) | |
5 | 4 | elv 3502 | . . . . . 6 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥)) |
6 | brcnvg 5753 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) | |
7 | 6 | el2v 3504 | . . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
8 | 7 | anbi2i 624 | . . . . . 6 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) |
9 | 5, 8 | bitri 277 | . . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) |
10 | 3, 9, 7 | 3bitr3i 303 | . . . 4 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦) |
11 | 10 | mobii 2630 | . . 3 ⊢ (∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦) |
12 | 1, 11 | bitri 277 | . 2 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦) |
13 | 12 | albii 1819 | 1 ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1534 ∈ wcel 2113 ∃*wmo 2619 ∃*wrmo 3144 Vcvv 3497 class class class wbr 5069 ◡ccnv 5557 ran crn 5559 ↾ cres 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rmo 3149 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 |
This theorem is referenced by: ineccnvmo2 35618 |
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