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 6593 | . . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐶 → 𝐹:𝐴⟶𝐶) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
4 | 3 | ffvelrnda 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶) |
5 | foelrn 6875 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | |
6 | 1, 5 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
7 | iunrdx.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) | |
8 | 7 | eleq2d 2901 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵)) |
9 | 4, 6, 8 | rexxfrd 5313 | . . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) |
10 | 9 | bicomd 225 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)) |
11 | 10 | abbidv 2888 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷}) |
12 | df-iun 4924 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
13 | df-iun 4924 | . 2 ⊢ ∪ 𝑦 ∈ 𝐶 𝐷 = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷} | |
14 | 11, 12, 13 | 3eqtr4g 2884 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {cab 2802 ∃wrex 3142 ∪ ciun 4922 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 |
This theorem is referenced by: volmeas 31494 |
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