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Mirrors > Home > MPE Home > Th. List > Mathboxes > inecmo | Structured version Visualization version GIF version |
Description: Lemma for ~? dfeldisj5 (via inecmo2 34121), ~? dfdisjs5 , ~? dfdisjALTV5 , ~? eldisjs5 (via inecmo3 34126, ~? cosscnvssid5 ), ~? dffunsALTV5 (via ineccnvmo 34122, ineccnvmo2 34125), and ~? dffunALTV5 (via ~? cossssid5 ). (Contributed by Peter Mazsa, 29-May-2018.) |
Ref | Expression |
---|---|
inecmo.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
inecmo | ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relelec 7787 | . . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑧)) | |
2 | relelec 7787 | . . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅 ↔ 𝐶𝑅𝑧)) | |
3 | 1, 2 | anbi12d 747 | . . . . . 6 ⊢ (Rel 𝑅 → ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) ↔ (𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧))) |
4 | 3 | imbi1d 331 | . . . . 5 ⊢ (Rel 𝑅 → (((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦))) |
5 | 4 | 2ralbidv 2989 | . . . 4 ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦))) |
6 | inecmo.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
7 | 6 | breq1d 4663 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝑧 ↔ 𝐶𝑅𝑧)) |
8 | 7 | rmo4 3399 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦)) |
9 | 5, 8 | syl6rbbr 279 | . . 3 ⊢ (Rel 𝑅 → (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))) |
10 | 9 | albidv 1849 | . 2 ⊢ (Rel 𝑅 → (∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))) |
11 | ineleq 34119 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)) | |
12 | ralcom4 3224 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)) | |
13 | 11, 12 | bitri 264 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)) |
14 | 10, 13 | syl6rbbr 279 | 1 ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃*wrmo 2915 ∩ cin 3573 ∅c0 3915 class class class wbr 4653 Rel wrel 5119 [cec 7740 |
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-sbc 3436 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 df-ima 5127 df-ec 7744 |
This theorem is referenced by: inecmo2 34121 ineccnvmo 34122 |
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