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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunrdx | Structured version Visualization version GIF version |
Description: Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
Ref | Expression |
---|---|
iunrdx.1 | ⊢ (𝜑 → 𝐹:𝐴–onto→𝐶) |
iunrdx.2 | ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
iunrdx | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunrdx.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐶) | |
2 | fof 6804 | . . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐶 → 𝐹:𝐴⟶𝐶) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
4 | 3 | ffvelcdmda 7085 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶) |
5 | foelrn 7107 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | |
6 | 1, 5 | sylan 578 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
7 | iunrdx.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) | |
8 | 7 | eleq2d 2817 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵)) |
9 | 4, 6, 8 | rexxfrd 5406 | . . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) |
10 | 9 | bicomd 222 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)) |
11 | 10 | abbidv 2799 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷}) |
12 | df-iun 4998 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
13 | df-iun 4998 | . 2 ⊢ ∪ 𝑦 ∈ 𝐶 𝐷 = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷} | |
14 | 11, 12, 13 | 3eqtr4g 2795 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {cab 2707 ∃wrex 3068 ∪ ciun 4996 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 |
This theorem is referenced by: volmeas 33527 |
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