| 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 6820 | . . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐶 → 𝐹:𝐴⟶𝐶) | |
| 3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
| 4 | 3 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶) |
| 5 | foelrn 7127 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | |
| 6 | 1, 5 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
| 7 | iunrdx.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) | |
| 8 | 7 | eleq2d 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵)) |
| 9 | 4, 6, 8 | rexxfrd 5409 | . . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) |
| 10 | 9 | bicomd 223 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)) |
| 11 | 10 | abbidv 2808 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷}) |
| 12 | df-iun 4993 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
| 13 | df-iun 4993 | . 2 ⊢ ∪ 𝑦 ∈ 𝐶 𝐷 = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷} | |
| 14 | 11, 12, 13 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ∪ ciun 4991 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 |
| This theorem is referenced by: volmeas 34232 |
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