Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjrdx | Structured version Visualization version GIF version |
Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
Ref | Expression |
---|---|
disjrdx.1 | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶) |
disjrdx.2 | ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
disjrdx | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjrdx.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶) | |
2 | f1of 6609 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐶 → 𝐹:𝐴⟶𝐶) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
4 | 3 | ffvelrnda 6845 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶) |
5 | f1ofveu 7145 | . . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) | |
6 | 1, 5 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) |
7 | eqcom 2828 | . . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥)) | |
8 | 7 | reubii 3391 | . . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
9 | 6, 8 | sylib 220 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
10 | disjrdx.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) | |
11 | 10 | eleq2d 2898 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵)) |
12 | 4, 9, 11 | rmoxfrd 30251 | . . . 4 ⊢ (𝜑 → (∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) |
13 | 12 | bicomd 225 | . . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)) |
14 | 13 | albidv 1917 | . 2 ⊢ (𝜑 → (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)) |
15 | df-disj 5024 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
16 | df-disj 5024 | . 2 ⊢ (Disj 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷) | |
17 | 14, 15, 16 | 3bitr4g 316 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 ∃*wrmo 3141 Disj wdisj 5023 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-disj 5024 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: tocyccntz 30781 volmeas 31485 carsggect 31571 |
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