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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 6801 | . . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐶 → 𝐹:𝐴⟶𝐶) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
4 | 3 | ffvelcdmda 7081 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶) |
5 | foelrn 7102 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | |
6 | 1, 5 | sylan 581 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
7 | iunrdx.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) | |
8 | 7 | eleq2d 2820 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵)) |
9 | 4, 6, 8 | rexxfrd 5405 | . . . 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 2802 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷}) |
12 | df-iun 4997 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
13 | df-iun 4997 | . 2 ⊢ ∪ 𝑦 ∈ 𝐶 𝐷 = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷} | |
14 | 11, 12, 13 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 ∪ ciun 4995 ⟶wf 6535 –onto→wfo 6537 ‘cfv 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-fo 6545 df-fv 6547 |
This theorem is referenced by: volmeas 33166 |
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