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Mirrors > Home > MPE Home > Th. List > Mathboxes > alrmomo | Structured version Visualization version GIF version |
Description: Lemma for ineccnvmo2 34125. (Contributed by Peter Mazsa, 3-Sep-2021.) |
Ref | Expression |
---|---|
alrmomo | ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 2920 | . . 3 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) | |
2 | cnvresrn 34116 | . . . . . 6 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | |
3 | 2 | breqi 4659 | . . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ 𝑦◡𝑅𝑥) |
4 | brresALTV 34032 | . . . . . . 7 ⊢ (𝑥 ∈ V → (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥))) | |
5 | 4 | elv 33983 | . . . . . 6 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥)) |
6 | brcnvg 5303 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) | |
7 | 6 | el2v 33984 | . . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
8 | 7 | anbi2i 730 | . . . . . 6 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) |
9 | 5, 8 | bitri 264 | . . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦)) |
10 | 3, 9, 7 | 3bitr3i 290 | . . . 4 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦) |
11 | 10 | mobii 2493 | . . 3 ⊢ (∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦) |
12 | 1, 11 | bitri 264 | . 2 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦) |
13 | 12 | albii 1747 | 1 ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∃*wmo 2471 ∃*wrmo 2915 Vcvv 3200 class class class wbr 4653 ◡ccnv 5113 ran crn 5115 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rmo 2920 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 |
This theorem is referenced by: ineccnvmo2 34125 |
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