| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > iundomg | Structured version Visualization version GIF version | ||
| Description: An upper bound for the cardinality of an indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.) |
| Ref | Expression |
|---|---|
| iunfo.1 | ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| iundomg.2 | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴) |
| iundomg.3 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) |
| iundomg.4 | ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) |
| Ref | Expression |
|---|---|
| iundomg | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iunfo.1 | . . . . 5 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
| 2 | iundomg.2 | . . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴) | |
| 3 | iundomg.3 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) | |
| 4 | 1, 2, 3 | iundom2g 10580 | . . . 4 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶)) |
| 5 | iundomg.4 | . . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) | |
| 6 | acndom2 10094 | . . . 4 ⊢ (𝑇 ≼ (𝐴 × 𝐶) → ((𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑇 ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)) | |
| 7 | 4, 5, 6 | sylc 65 | . . 3 ⊢ (𝜑 → 𝑇 ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) |
| 8 | 1 | iunfo 10579 | . . 3 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 |
| 9 | fodomacn 10096 | . . 3 ⊢ (𝑇 ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵 → ((2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇)) | |
| 10 | 7, 8, 9 | mpisyl 21 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇) |
| 11 | domtr 9047 | . 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇 ∧ 𝑇 ≼ (𝐴 × 𝐶)) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) | |
| 12 | 10, 4, 11 | syl2anc 584 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 ∪ ciun 4991 class class class wbr 5143 × cxp 5683 ↾ cres 5687 –onto→wfo 6559 (class class class)co 7431 2nd c2nd 8013 ↑m cmap 8866 ≼ cdom 8983 AC wacn 9978 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 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-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-dom 8987 df-acn 9982 |
| This theorem is referenced by: iundom 10582 iunctb 10614 |
| Copyright terms: Public domain | W3C validator |