| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > founiiun | Structured version Visualization version GIF version | ||
| Description: Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| founiiun | ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniiun 5058 | . 2 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦 | |
| 2 | foelcdmi 6970 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) | |
| 3 | eqimss2 4043 | . . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥)) | |
| 4 | 3 | reximi 3084 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥)) |
| 5 | 2, 4 | syl 17 | . . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥)) |
| 6 | 5 | ralrimiva 3146 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥)) |
| 7 | iunss2 5049 | . . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) | |
| 8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) |
| 9 | fof 6820 | . . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
| 10 | 9 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
| 11 | ssidd 4007 | . . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) | |
| 12 | sseq2 4010 | . . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥))) | |
| 13 | 12 | rspcev 3622 | . . . . . 6 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦) |
| 14 | 10, 11, 13 | syl2anc 584 | . . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦) |
| 15 | 14 | ralrimiva 3146 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦) |
| 16 | iunss2 5049 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦) | |
| 17 | 15, 16 | syl 17 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦) |
| 18 | 8, 17 | eqssd 4001 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) |
| 19 | 1, 18 | eqtrid 2789 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∪ cuni 4907 ∪ ciun 4991 –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: founiiun0 45195 issalnnd 46360 caragenunicl 46539 isomenndlem 46545 |
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